L(s) = 1 | − 2-s + 4-s + 7-s − 8-s + 5·13-s − 14-s + 16-s + 17-s − 4·19-s − 3·23-s − 5·25-s − 5·26-s + 28-s + 9·29-s − 31-s − 32-s − 34-s − 4·37-s + 4·38-s + 9·41-s − 43-s + 3·46-s − 6·47-s + 49-s + 5·50-s + 5·52-s − 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.625·23-s − 25-s − 0.980·26-s + 0.188·28-s + 1.67·29-s − 0.179·31-s − 0.176·32-s − 0.171·34-s − 0.657·37-s + 0.648·38-s + 1.40·41-s − 0.152·43-s + 0.442·46-s − 0.875·47-s + 1/7·49-s + 0.707·50-s + 0.693·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.80504351037351, −12.67517459118911, −11.96543710090604, −11.65021353330150, −11.05315934308307, −10.72453412813259, −10.39443392254165, −9.720662494502979, −9.387925178918053, −8.777167391637747, −8.347499437316864, −7.986776865090434, −7.698218581422162, −6.765806651001744, −6.557916907114458, −5.973271951338873, −5.636280842177224, −4.813291010415901, −4.353823880311310, −3.734822165604782, −3.279393855096387, −2.536227630374186, −1.951736702071462, −1.426516177651073, −0.8113551302299991, 0,
0.8113551302299991, 1.426516177651073, 1.951736702071462, 2.536227630374186, 3.279393855096387, 3.734822165604782, 4.353823880311310, 4.813291010415901, 5.636280842177224, 5.973271951338873, 6.557916907114458, 6.765806651001744, 7.698218581422162, 7.986776865090434, 8.347499437316864, 8.777167391637747, 9.387925178918053, 9.720662494502979, 10.39443392254165, 10.72453412813259, 11.05315934308307, 11.65021353330150, 11.96543710090604, 12.67517459118911, 12.80504351037351