| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s − 3·7-s + 8-s − 2·9-s + 2·10-s − 12-s + 2·13-s − 3·14-s − 2·15-s + 16-s + 2·17-s − 2·18-s − 19-s + 2·20-s + 3·21-s − 23-s − 24-s − 25-s + 2·26-s + 5·27-s − 3·28-s − 5·29-s − 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.288·12-s + 0.554·13-s − 0.801·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s − 0.566·28-s − 0.928·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.78151218288077478977168753185, −6.93584518468912270900995746603, −6.15364013153051301443253467181, −5.84051037869200021796884427587, −5.30812815840586472937784833666, −4.16020356360102787822054571441, −3.35469611081304078184838017364, −2.59902861333584776243201559533, −1.52581543186035418373331338991, 0,
1.52581543186035418373331338991, 2.59902861333584776243201559533, 3.35469611081304078184838017364, 4.16020356360102787822054571441, 5.30812815840586472937784833666, 5.84051037869200021796884427587, 6.15364013153051301443253467181, 6.93584518468912270900995746603, 7.78151218288077478977168753185
