| L(s) = 1 | − 4·2-s − 6·3-s + 12·4-s + 10·5-s + 24·6-s − 32·8-s + 27·9-s − 40·10-s − 58·11-s − 72·12-s − 56·13-s − 60·15-s + 80·16-s + 10·17-s − 108·18-s + 2·19-s + 120·20-s + 232·22-s − 130·23-s + 192·24-s + 75·25-s + 224·26-s − 108·27-s − 34·29-s + 240·30-s − 344·31-s − 192·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s − 1.58·11-s − 1.73·12-s − 1.19·13-s − 1.03·15-s + 5/4·16-s + 0.142·17-s − 1.41·18-s + 0.0241·19-s + 1.34·20-s + 2.24·22-s − 1.17·23-s + 1.63·24-s + 3/5·25-s + 1.68·26-s − 0.769·27-s − 0.217·29-s + 1.46·30-s − 1.99·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4290803630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4290803630\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 + 58 T + 2998 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 56 T + 3158 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T - 2774 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 1094 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 130 T + 15934 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 34 T - 12038 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 344 T + 70986 T^{2} + 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 54 T - 43910 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 60 T + 130662 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 206 T + 84278 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 958 T + 432542 T^{2} - 958 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1350 T + 752874 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 132 T + 364614 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 392938 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1486 T + 1149030 T^{2} - 1486 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1364 T + 1130446 T^{2} + 1364 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 44 T + 382598 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 1892 T + 2020310 T^{2} + 1892 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 596 T + 1369462 T^{2} + 596 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 508 T + 524342 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.336966699081090660758272621256, −9.123346523768956976957594864025, −8.496197626730423396567674362174, −8.250193898491134162397311979532, −7.62583047494636630813710964624, −7.26505599238655765865993377651, −7.05085359196461389134217533321, −6.76844867471864814739238679126, −5.80343602249349658517072269163, −5.75382183712185734036917071533, −5.39375232189804899949338198421, −5.18635379776903935563649310080, −4.18791752993274801362083489573, −3.96068395772349139824160139283, −2.89163286686456329481244948126, −2.51764763466013534007012884497, −2.04646877574156253612592973666, −1.60824305236788146370921990300, −0.71912833505289888278488148277, −0.28236086077963865199454561622,
0.28236086077963865199454561622, 0.71912833505289888278488148277, 1.60824305236788146370921990300, 2.04646877574156253612592973666, 2.51764763466013534007012884497, 2.89163286686456329481244948126, 3.96068395772349139824160139283, 4.18791752993274801362083489573, 5.18635379776903935563649310080, 5.39375232189804899949338198421, 5.75382183712185734036917071533, 5.80343602249349658517072269163, 6.76844867471864814739238679126, 7.05085359196461389134217533321, 7.26505599238655765865993377651, 7.62583047494636630813710964624, 8.250193898491134162397311979532, 8.496197626730423396567674362174, 9.123346523768956976957594864025, 9.336966699081090660758272621256
