| L(s) = 1 | − 1.31·3-s − 5-s + 1.52·7-s − 1.28·9-s − 1.09·11-s − 0.836·13-s + 1.31·15-s + 2·17-s + 19-s − 1.99·21-s − 1.52·23-s + 25-s + 5.61·27-s − 4.23·29-s + 5.18·31-s + 1.43·33-s − 1.52·35-s − 2.83·37-s + 1.09·39-s + 10.2·41-s − 11.4·43-s + 1.28·45-s + 1.52·47-s − 4.67·49-s − 2.62·51-s + 6.83·53-s + 1.09·55-s + ⋯ |
| L(s) = 1 | − 0.756·3-s − 0.447·5-s + 0.576·7-s − 0.426·9-s − 0.330·11-s − 0.232·13-s + 0.338·15-s + 0.485·17-s + 0.229·19-s − 0.436·21-s − 0.318·23-s + 0.200·25-s + 1.08·27-s − 0.786·29-s + 0.931·31-s + 0.250·33-s − 0.257·35-s − 0.466·37-s + 0.175·39-s + 1.59·41-s − 1.74·43-s + 0.190·45-s + 0.222·47-s − 0.667·49-s − 0.367·51-s + 0.939·53-s + 0.147·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.31T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 1.09T + 11T^{2} \) |
| 13 | \( 1 + 0.836T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 1.52T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 5.18T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.52T + 47T^{2} \) |
| 53 | \( 1 - 6.83T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 + 8.66T + 79T^{2} \) |
| 83 | \( 1 - 7.19T + 83T^{2} \) |
| 89 | \( 1 + 8.23T + 89T^{2} \) |
| 97 | \( 1 + 3.39T + 97T^{2} \) |
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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
−7.82286000404054031369916395494, −6.98100430878597522383864785830, −6.24522792206782373099169042844, −5.44295474024535211009573834635, −5.01505391906774734952464752012, −4.15683079864013331790346314076, −3.24631044828590750152350328440, −2.32746354685020143711793545997, −1.11880242434552781746461996068, 0,
1.11880242434552781746461996068, 2.32746354685020143711793545997, 3.24631044828590750152350328440, 4.15683079864013331790346314076, 5.01505391906774734952464752012, 5.44295474024535211009573834635, 6.24522792206782373099169042844, 6.98100430878597522383864785830, 7.82286000404054031369916395494
