| L(s) = 1 | + 2-s + 3·3-s − 4-s + 3·6-s + 2·7-s − 8-s + 6·9-s − 8·11-s − 3·12-s + 4·13-s + 2·14-s − 16-s − 2·17-s + 6·18-s − 14·19-s + 6·21-s − 8·22-s − 12·23-s − 3·24-s + 4·26-s + 10·27-s − 2·28-s − 16·29-s − 8·31-s − 32-s − 24·33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s − 1/2·4-s + 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s − 2.41·11-s − 0.866·12-s + 1.10·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 1.41·18-s − 3.21·19-s + 1.30·21-s − 1.70·22-s − 2.50·23-s − 0.612·24-s + 0.784·26-s + 1.92·27-s − 0.377·28-s − 2.97·29-s − 1.43·31-s − 0.176·32-s − 4.17·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 59^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 59 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} + 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 8 T + 30 T^{2} + 84 T^{3} + 30 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 28 T^{2} - 106 T^{3} + 28 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 2 T + 12 T^{2} - 18 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 14 T + 106 T^{2} + 536 T^{3} + 106 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 12 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 16 T + 165 T^{2} + 36 p T^{3} + 165 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 85 T^{2} + 464 T^{3} + 85 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 12 T + 152 T^{2} + 926 T^{3} + 152 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 - 14 T + 162 T^{2} - 1220 T^{3} + 162 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 64 T^{2} + 210 T^{3} + 64 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 4 T - 3 T^{2} + 428 T^{3} - 3 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 22 T + 258 T^{2} - 2260 T^{3} + 258 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 189 T^{2} + 504 T^{3} + 189 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 4 p T^{2} + 2978 T^{3} + 4 p^{2} T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 30 T + 530 T^{2} + 5668 T^{3} + 530 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 28 T + 413 T^{2} - 4184 T^{3} + 413 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 60 T^{2} + 606 T^{3} + 60 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 131 T^{2} + 1180 T^{3} + 131 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81686478127053212427748629476, −7.72940578088587857528520008585, −7.26366337588638904633681014584, −7.16706097197175407132358473110, −6.85848451845346807744438024661, −6.63797753054303032369421883465, −6.16929999811151830568756194418, −5.90000664313929326436980341563, −5.79124578622122242096240182702, −5.53703435190987716221245208008, −5.03059668575257942714310428600, −4.95081616734909176767499100148, −4.88644672261356172907759161212, −4.22553637409131047300253931490, −4.10385972751975505092820958840, −3.92995026326766255800611140248, −3.76028593739826420236730409484, −3.57746037743026133578180986988, −3.14439100027274566842293534964, −2.67312644107249319047653755531, −2.30757395701807134839833320283, −2.15408870724119032451571935162, −2.04426037171639293348590319013, −1.64539912015820128089613438444, −1.42572051857583430553350051375, 0, 0, 0,
1.42572051857583430553350051375, 1.64539912015820128089613438444, 2.04426037171639293348590319013, 2.15408870724119032451571935162, 2.30757395701807134839833320283, 2.67312644107249319047653755531, 3.14439100027274566842293534964, 3.57746037743026133578180986988, 3.76028593739826420236730409484, 3.92995026326766255800611140248, 4.10385972751975505092820958840, 4.22553637409131047300253931490, 4.88644672261356172907759161212, 4.95081616734909176767499100148, 5.03059668575257942714310428600, 5.53703435190987716221245208008, 5.79124578622122242096240182702, 5.90000664313929326436980341563, 6.16929999811151830568756194418, 6.63797753054303032369421883465, 6.85848451845346807744438024661, 7.16706097197175407132358473110, 7.26366337588638904633681014584, 7.72940578088587857528520008585, 7.81686478127053212427748629476
