L(s) = 1 | − 2·2-s + 2·3-s + 4·4-s − 4·6-s − 7·7-s − 8·8-s − 23·9-s − 27·11-s + 8·12-s + 64·13-s + 14·14-s + 16·16-s + 24·17-s + 46·18-s + 62·19-s − 14·21-s + 54·22-s + 105·23-s − 16·24-s − 128·26-s − 100·27-s − 28·28-s + 141·29-s − 124·31-s − 32·32-s − 54·33-s − 48·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.384·3-s + 1/2·4-s − 0.272·6-s − 0.377·7-s − 0.353·8-s − 0.851·9-s − 0.740·11-s + 0.192·12-s + 1.36·13-s + 0.267·14-s + 1/4·16-s + 0.342·17-s + 0.602·18-s + 0.748·19-s − 0.145·21-s + 0.523·22-s + 0.951·23-s − 0.136·24-s − 0.965·26-s − 0.712·27-s − 0.188·28-s + 0.902·29-s − 0.718·31-s − 0.176·32-s − 0.284·33-s − 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.367097560\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.367097560\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 27 T + p^{3} T^{2} \) |
| 13 | \( 1 - 64 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 62 T + p^{3} T^{2} \) |
| 23 | \( 1 - 105 T + p^{3} T^{2} \) |
| 29 | \( 1 - 141 T + p^{3} T^{2} \) |
| 31 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 439 T + p^{3} T^{2} \) |
| 41 | \( 1 + 354 T + p^{3} T^{2} \) |
| 43 | \( 1 - 211 T + p^{3} T^{2} \) |
| 47 | \( 1 - 102 T + p^{3} T^{2} \) |
| 53 | \( 1 - 306 T + p^{3} T^{2} \) |
| 59 | \( 1 - 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 410 T + p^{3} T^{2} \) |
| 67 | \( 1 - 349 T + p^{3} T^{2} \) |
| 71 | \( 1 + 339 T + p^{3} T^{2} \) |
| 73 | \( 1 - 70 T + p^{3} T^{2} \) |
| 79 | \( 1 - 731 T + p^{3} T^{2} \) |
| 83 | \( 1 + 528 T + p^{3} T^{2} \) |
| 89 | \( 1 - 960 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1340 T + p^{3} T^{2} \) |
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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
−10.99337076074995362694527464545, −10.05737658074226236431093792009, −9.064981121128506360375738848594, −8.393428283614674170225810767068, −7.50805508863986077707979634556, −6.30800414976186257160673741172, −5.35998193112450924891206231201, −3.54971603366328936255168652454, −2.59161560455248369418694824516, −0.863195869564064719933081326343,
0.863195869564064719933081326343, 2.59161560455248369418694824516, 3.54971603366328936255168652454, 5.35998193112450924891206231201, 6.30800414976186257160673741172, 7.50805508863986077707979634556, 8.393428283614674170225810767068, 9.064981121128506360375738848594, 10.05737658074226236431093792009, 10.99337076074995362694527464545