L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 11-s − 13-s − 14-s + 16-s + 3·17-s + 5·19-s − 22-s + 23-s − 5·25-s + 26-s + 28-s − 9·29-s + 5·31-s − 32-s − 3·34-s + 2·37-s − 5·38-s − 10·43-s + 44-s − 46-s − 12·47-s + 49-s + 5·50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 0.301·11-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.14·19-s − 0.213·22-s + 0.208·23-s − 25-s + 0.196·26-s + 0.188·28-s − 1.67·29-s + 0.898·31-s − 0.176·32-s − 0.514·34-s + 0.328·37-s − 0.811·38-s − 1.52·43-s + 0.150·44-s − 0.147·46-s − 1.75·47-s + 1/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31878 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31878 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.33434463877780, −14.79370915551591, −14.46903544249888, −13.71212679312148, −13.28298357957697, −12.59591320952039, −11.93030166898837, −11.49045355374088, −11.26056157933178, −10.35625631421181, −9.747000806403647, −9.647385311486962, −8.848394267725194, −8.218937493518383, −7.692617647564697, −7.363017886584984, −6.541847259299524, −6.031993520973335, −5.221520112738175, −4.879368113147502, −3.725319699273639, −3.401162101014293, −2.442214580606301, −1.706991874593424, −1.054092041234294, 0,
1.054092041234294, 1.706991874593424, 2.442214580606301, 3.401162101014293, 3.725319699273639, 4.879368113147502, 5.221520112738175, 6.031993520973335, 6.541847259299524, 7.363017886584984, 7.692617647564697, 8.218937493518383, 8.848394267725194, 9.647385311486962, 9.747000806403647, 10.35625631421181, 11.26056157933178, 11.49045355374088, 11.93030166898837, 12.59591320952039, 13.28298357957697, 13.71212679312148, 14.46903544249888, 14.79370915551591, 15.33434463877780