| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s + 3·9-s + 10-s − 4·11-s + 5·13-s − 2·14-s − 16-s − 3·18-s − 19-s + 4·22-s − 23-s + 5·25-s − 5·26-s − 6·29-s + 8·31-s − 2·35-s − 8·37-s + 38-s − 40-s + 2·41-s + 8·43-s − 3·45-s + 46-s + 3·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 0.755·7-s + 0.353·8-s + 9-s + 0.316·10-s − 1.20·11-s + 1.38·13-s − 0.534·14-s − 1/4·16-s − 0.707·18-s − 0.229·19-s + 0.852·22-s − 0.208·23-s + 25-s − 0.980·26-s − 1.11·29-s + 1.43·31-s − 0.338·35-s − 1.31·37-s + 0.162·38-s − 0.158·40-s + 0.312·41-s + 1.21·43-s − 0.447·45-s + 0.147·46-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70756 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014233297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014233297\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−12.11348848711017560674869479190, −11.61207433860716108088709370398, −11.00043173241203870047995582953, −10.75719014192170363395181969814, −10.37755949440868859479027285281, −9.897344791232310072053197966894, −9.216148772297698605205221714984, −8.799581155032277111160580789939, −8.134742003688633062104702807013, −8.082498302792329812782963424459, −7.43174728823283763587912723063, −6.89769889116600464815504637830, −6.36111339723882760109540933523, −5.41463548359761481081682097917, −5.14473630358566024827421205906, −4.18869598427721939322514680335, −3.96786597316900554074131776397, −2.89827637400626757099685399685, −1.93039725822083227889206270090, −0.965864172298663692739533721417,
0.965864172298663692739533721417, 1.93039725822083227889206270090, 2.89827637400626757099685399685, 3.96786597316900554074131776397, 4.18869598427721939322514680335, 5.14473630358566024827421205906, 5.41463548359761481081682097917, 6.36111339723882760109540933523, 6.89769889116600464815504637830, 7.43174728823283763587912723063, 8.082498302792329812782963424459, 8.134742003688633062104702807013, 8.799581155032277111160580789939, 9.216148772297698605205221714984, 9.897344791232310072053197966894, 10.37755949440868859479027285281, 10.75719014192170363395181969814, 11.00043173241203870047995582953, 11.61207433860716108088709370398, 12.11348848711017560674869479190
