| L(s) = 1 | + 2·2-s + 3-s + 4·4-s − 5·5-s + 2·6-s − 18·7-s + 8·8-s − 26·9-s − 10·10-s − 32·11-s + 4·12-s − 47·13-s − 36·14-s − 5·15-s + 16·16-s + 20·17-s − 52·18-s + 36·19-s − 20·20-s − 18·21-s − 64·22-s − 23·23-s + 8·24-s + 25·25-s − 94·26-s − 53·27-s − 72·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.192·3-s + 1/2·4-s − 0.447·5-s + 0.136·6-s − 0.971·7-s + 0.353·8-s − 0.962·9-s − 0.316·10-s − 0.877·11-s + 0.0962·12-s − 1.00·13-s − 0.687·14-s − 0.0860·15-s + 1/4·16-s + 0.285·17-s − 0.680·18-s + 0.434·19-s − 0.223·20-s − 0.187·21-s − 0.620·22-s − 0.208·23-s + 0.0680·24-s + 1/5·25-s − 0.709·26-s − 0.377·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 20 T + p^{3} T^{2} \) |
| 19 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 27 T + p^{3} T^{2} \) |
| 31 | \( 1 + 33 T + p^{3} T^{2} \) |
| 37 | \( 1 - 56 T + p^{3} T^{2} \) |
| 41 | \( 1 + 157 T + p^{3} T^{2} \) |
| 43 | \( 1 - 18 T + p^{3} T^{2} \) |
| 47 | \( 1 - 65 T + p^{3} T^{2} \) |
| 53 | \( 1 + 14 T + p^{3} T^{2} \) |
| 59 | \( 1 + 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 552 T + p^{3} T^{2} \) |
| 67 | \( 1 + 156 T + p^{3} T^{2} \) |
| 71 | \( 1 - 699 T + p^{3} T^{2} \) |
| 73 | \( 1 + 609 T + p^{3} T^{2} \) |
| 79 | \( 1 + 644 T + p^{3} T^{2} \) |
| 83 | \( 1 - 512 T + p^{3} T^{2} \) |
| 89 | \( 1 + 102 T + p^{3} T^{2} \) |
| 97 | \( 1 - 578 T + p^{3} T^{2} \) |
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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
−11.50429916990015101305589071589, −10.39818643003529938431121028697, −9.414598983480301621781848422522, −8.107853789280623336959830261582, −7.19178629219405417593487885588, −5.96236358588767764811469126611, −4.95900924498192662948316038541, −3.46320835416819586200571433587, −2.58546977397398571614001380484, 0,
2.58546977397398571614001380484, 3.46320835416819586200571433587, 4.95900924498192662948316038541, 5.96236358588767764811469126611, 7.19178629219405417593487885588, 8.107853789280623336959830261582, 9.414598983480301621781848422522, 10.39818643003529938431121028697, 11.50429916990015101305589071589
