L(s) = 1 | − 2·2-s + 4·4-s + 7·7-s − 8·8-s + 42·11-s + 20·13-s − 14·14-s + 16·16-s + 96·17-s + 19·19-s − 84·22-s − 102·23-s − 125·25-s − 40·26-s + 28·28-s − 90·29-s − 196·31-s − 32·32-s − 192·34-s − 214·37-s − 38·38-s − 378·41-s − 376·43-s + 168·44-s + 204·46-s − 216·47-s + 49·49-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s + 1.15·11-s + 0.426·13-s − 0.267·14-s + 1/4·16-s + 1.36·17-s + 0.229·19-s − 0.814·22-s − 0.924·23-s − 25-s − 0.301·26-s + 0.188·28-s − 0.576·29-s − 1.13·31-s − 0.176·32-s − 0.968·34-s − 0.950·37-s − 0.162·38-s − 1.43·41-s − 1.33·43-s + 0.575·44-s + 0.653·46-s − 0.670·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
good | 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 - 96 T + p^{3} T^{2} \) |
| 23 | \( 1 + 102 T + p^{3} T^{2} \) |
| 29 | \( 1 + 90 T + p^{3} T^{2} \) |
| 31 | \( 1 + 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 216 T + p^{3} T^{2} \) |
| 53 | \( 1 - 750 T + p^{3} T^{2} \) |
| 59 | \( 1 - 252 T + p^{3} T^{2} \) |
| 61 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 + 286 T + p^{3} T^{2} \) |
| 71 | \( 1 + 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 358 T + p^{3} T^{2} \) |
| 79 | \( 1 + 862 T + p^{3} T^{2} \) |
| 83 | \( 1 - 384 T + p^{3} T^{2} \) |
| 89 | \( 1 + 42 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1240 T + p^{3} T^{2} \) |
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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
−8.359921277765625592079458920275, −7.52382748968691349419215861011, −6.87454452489058883355933374924, −5.92426203209233700350860247174, −5.29567461679804919600412753989, −3.93302385365115743982798220215, −3.39160570846904907844243978831, −1.90407638539730316963381596749, −1.31418824217718510392644251921, 0,
1.31418824217718510392644251921, 1.90407638539730316963381596749, 3.39160570846904907844243978831, 3.93302385365115743982798220215, 5.29567461679804919600412753989, 5.92426203209233700350860247174, 6.87454452489058883355933374924, 7.52382748968691349419215861011, 8.359921277765625592079458920275