| L(s) = 1 | + 4·2-s + 23·3-s + 16·4-s + 92·6-s − 49·7-s + 64·8-s + 286·9-s + 555·11-s + 368·12-s + 241·13-s − 196·14-s + 256·16-s + 1.49e3·17-s + 1.14e3·18-s − 2.03e3·19-s − 1.12e3·21-s + 2.22e3·22-s + 1.23e3·23-s + 1.47e3·24-s + 964·26-s + 989·27-s − 784·28-s − 5.00e3·29-s + 5.69e3·31-s + 1.02e3·32-s + 1.27e4·33-s + 5.96e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.47·3-s + 1/2·4-s + 1.04·6-s − 0.377·7-s + 0.353·8-s + 1.17·9-s + 1.38·11-s + 0.737·12-s + 0.395·13-s − 0.267·14-s + 1/4·16-s + 1.25·17-s + 0.832·18-s − 1.29·19-s − 0.557·21-s + 0.977·22-s + 0.484·23-s + 0.521·24-s + 0.279·26-s + 0.261·27-s − 0.188·28-s − 1.10·29-s + 1.06·31-s + 0.176·32-s + 2.04·33-s + 0.884·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(6.228591284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.228591284\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 23 T + p^{5} T^{2} \) |
| 11 | \( 1 - 555 T + p^{5} T^{2} \) |
| 13 | \( 1 - 241 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1491 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2038 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1230 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5001 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5696 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5602 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2424 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 23163 T + p^{5} T^{2} \) |
| 53 | \( 1 - 25296 T + p^{5} T^{2} \) |
| 59 | \( 1 - 5724 T + p^{5} T^{2} \) |
| 61 | \( 1 + 592 p T + p^{5} T^{2} \) |
| 67 | \( 1 + 66104 T + p^{5} T^{2} \) |
| 71 | \( 1 - 16080 T + p^{5} T^{2} \) |
| 73 | \( 1 - 80482 T + p^{5} T^{2} \) |
| 79 | \( 1 + 64147 T + p^{5} T^{2} \) |
| 83 | \( 1 - 106284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71676 T + p^{5} T^{2} \) |
| 97 | \( 1 + 151025 T + p^{5} T^{2} \) |
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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
−10.61234125041451395106381602287, −9.530000301948082952898737052970, −8.824780846598125994908504529734, −7.86555224858842523835258628322, −6.84717888260662375872101724934, −5.85584036330747094706909429114, −4.21371955047430511806352741593, −3.56787053380616474245354600353, −2.53723454779012524080102462970, −1.30074561281391937363574372249,
1.30074561281391937363574372249, 2.53723454779012524080102462970, 3.56787053380616474245354600353, 4.21371955047430511806352741593, 5.85584036330747094706909429114, 6.84717888260662375872101724934, 7.86555224858842523835258628322, 8.824780846598125994908504529734, 9.530000301948082952898737052970, 10.61234125041451395106381602287
