| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 16-s + 18-s + 4·19-s − 6·22-s − 2·24-s − 5·25-s − 2·26-s + 4·27-s − 4·31-s + 32-s + 12·33-s + 36-s + 4·37-s + 4·38-s + 4·39-s + 6·41-s + 8·43-s − 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 1.27·22-s − 0.408·24-s − 25-s − 0.392·26-s + 0.769·27-s − 0.718·31-s + 0.176·32-s + 2.08·33-s + 1/6·36-s + 0.657·37-s + 0.648·38-s + 0.640·39-s + 0.937·41-s + 1.21·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.62409703535378, −15.04205630843256, −14.19917467229632, −14.00883159843997, −13.11601907067090, −12.80742795298601, −12.35698114775160, −11.69907118063491, −11.24841503617883, −10.83382041501906, −10.23877217507859, −9.767855378236961, −9.028329174338571, −8.060950817464079, −7.581014872933697, −7.235998692806199, −6.258703329131722, −5.836049490258861, −5.351566171254877, −4.904474632161383, −4.299976143335690, −3.363295403306189, −2.684192714115655, −2.070233304346154, −0.8623572496243492, 0,
0.8623572496243492, 2.070233304346154, 2.684192714115655, 3.363295403306189, 4.299976143335690, 4.904474632161383, 5.351566171254877, 5.836049490258861, 6.258703329131722, 7.235998692806199, 7.581014872933697, 8.060950817464079, 9.028329174338571, 9.767855378236961, 10.23877217507859, 10.83382041501906, 11.24841503617883, 11.69907118063491, 12.35698114775160, 12.80742795298601, 13.11601907067090, 14.00883159843997, 14.19917467229632, 15.04205630843256, 15.62409703535378
