| L(s) = 1 | − 2·2-s + 7·3-s + 4·4-s − 14·6-s − 8·8-s + 22·9-s − 33·11-s + 28·12-s − 43·13-s + 16·16-s + 111·17-s − 44·18-s + 70·19-s + 66·22-s − 42·23-s − 56·24-s + 86·26-s − 35·27-s − 225·29-s + 88·31-s − 32·32-s − 231·33-s − 222·34-s + 88·36-s + 34·37-s − 140·38-s − 301·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s − 0.952·6-s − 0.353·8-s + 0.814·9-s − 0.904·11-s + 0.673·12-s − 0.917·13-s + 1/4·16-s + 1.58·17-s − 0.576·18-s + 0.845·19-s + 0.639·22-s − 0.380·23-s − 0.476·24-s + 0.648·26-s − 0.249·27-s − 1.44·29-s + 0.509·31-s − 0.176·32-s − 1.21·33-s − 1.11·34-s + 0.407·36-s + 0.151·37-s − 0.597·38-s − 1.23·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 111 T + p^{3} T^{2} \) |
| 19 | \( 1 - 70 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 + 225 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 432 T + p^{3} T^{2} \) |
| 43 | \( 1 - 178 T + p^{3} T^{2} \) |
| 47 | \( 1 - 411 T + p^{3} T^{2} \) |
| 53 | \( 1 - 708 T + p^{3} T^{2} \) |
| 59 | \( 1 + 480 T + p^{3} T^{2} \) |
| 61 | \( 1 + 812 T + p^{3} T^{2} \) |
| 67 | \( 1 + 596 T + p^{3} T^{2} \) |
| 71 | \( 1 - 432 T + p^{3} T^{2} \) |
| 73 | \( 1 + 358 T + p^{3} T^{2} \) |
| 79 | \( 1 - 425 T + p^{3} T^{2} \) |
| 83 | \( 1 - 972 T + p^{3} T^{2} \) |
| 89 | \( 1 + 960 T + p^{3} T^{2} \) |
| 97 | \( 1 + 709 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.132478153244955523894196495604, −7.59953717994969246760970273533, −7.26782739448531188424600824142, −5.84933490963442357416741125131, −5.15167160801404927162349140227, −3.82699864112767742249983888943, −3.01328196962603800053854271789, −2.39117021852438224984434648468, −1.35653990563049898833965073803, 0,
1.35653990563049898833965073803, 2.39117021852438224984434648468, 3.01328196962603800053854271789, 3.82699864112767742249983888943, 5.15167160801404927162349140227, 5.84933490963442357416741125131, 7.26782739448531188424600824142, 7.59953717994969246760970273533, 8.132478153244955523894196495604
