| L(s) = 1 | − 2-s − 7·4-s − 14·5-s − 7·7-s + 15·8-s + 14·10-s − 47·11-s − 86·13-s + 7·14-s + 41·16-s − 9·17-s − 131·19-s + 98·20-s + 47·22-s − 12·23-s + 71·25-s + 86·26-s + 49·28-s − 260·29-s − 54·31-s − 161·32-s + 9·34-s + 98·35-s − 246·37-s + 131·38-s − 210·40-s + 383·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 1.25·5-s − 0.377·7-s + 0.662·8-s + 0.442·10-s − 1.28·11-s − 1.83·13-s + 0.133·14-s + 0.640·16-s − 0.128·17-s − 1.58·19-s + 1.09·20-s + 0.455·22-s − 0.108·23-s + 0.567·25-s + 0.648·26-s + 0.330·28-s − 1.66·29-s − 0.312·31-s − 0.889·32-s + 0.0453·34-s + 0.473·35-s − 1.09·37-s + 0.559·38-s − 0.830·40-s + 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 47 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 9 T + p^{3} T^{2} \) |
| 19 | \( 1 + 131 T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 260 T + p^{3} T^{2} \) |
| 31 | \( 1 + 54 T + p^{3} T^{2} \) |
| 37 | \( 1 + 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 383 T + p^{3} T^{2} \) |
| 43 | \( 1 + 169 T + p^{3} T^{2} \) |
| 47 | \( 1 - 96 T + p^{3} T^{2} \) |
| 53 | \( 1 - 300 T + p^{3} T^{2} \) |
| 59 | \( 1 - 429 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 + 155 T + p^{3} T^{2} \) |
| 71 | \( 1 - 72 T + p^{3} T^{2} \) |
| 73 | \( 1 - 117 T + p^{3} T^{2} \) |
| 79 | \( 1 + 526 T + p^{3} T^{2} \) |
| 83 | \( 1 - 576 T + p^{3} T^{2} \) |
| 89 | \( 1 + 278 T + p^{3} T^{2} \) |
| 97 | \( 1 + 201 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.442743830681247730759628175689, −8.502598045886930119835416055184, −7.71590223777863959295182677546, −7.17233186974220298853115080178, −5.50561701561228407187882647115, −4.60684721590440854948650657165, −3.77207709307496969488493918383, −2.36755685101964811805412793216, 0, 0,
2.36755685101964811805412793216, 3.77207709307496969488493918383, 4.60684721590440854948650657165, 5.50561701561228407187882647115, 7.17233186974220298853115080178, 7.71590223777863959295182677546, 8.502598045886930119835416055184, 9.442743830681247730759628175689
