| L(s) = 1 | − 2.27·2-s − 0.785·3-s + 3.15·4-s − 4.14·5-s + 1.78·6-s − 2.62·8-s − 2.38·9-s + 9.40·10-s + 11-s − 2.47·12-s + 13-s + 3.25·15-s − 0.345·16-s − 6.36·17-s + 5.41·18-s + 3.29·19-s − 13.0·20-s − 2.27·22-s − 3.01·23-s + 2.06·24-s + 12.1·25-s − 2.27·26-s + 4.22·27-s + 8.47·29-s − 7.38·30-s − 1.95·31-s + 6.04·32-s + ⋯ |
| L(s) = 1 | − 1.60·2-s − 0.453·3-s + 1.57·4-s − 1.85·5-s + 0.728·6-s − 0.929·8-s − 0.794·9-s + 2.97·10-s + 0.301·11-s − 0.715·12-s + 0.277·13-s + 0.839·15-s − 0.0862·16-s − 1.54·17-s + 1.27·18-s + 0.754·19-s − 2.92·20-s − 0.484·22-s − 0.629·23-s + 0.421·24-s + 2.43·25-s − 0.445·26-s + 0.813·27-s + 1.57·29-s − 1.34·30-s − 0.350·31-s + 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1283568299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1283568299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.27T + 2T^{2} \) |
| 3 | \( 1 + 0.785T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 17 | \( 1 + 6.36T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 - 0.328T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.32T + 43T^{2} \) |
| 47 | \( 1 + 6.69T + 47T^{2} \) |
| 53 | \( 1 + 2.78T + 53T^{2} \) |
| 59 | \( 1 - 5.58T + 59T^{2} \) |
| 61 | \( 1 + 3.90T + 61T^{2} \) |
| 67 | \( 1 - 5.13T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 + 9.64T + 89T^{2} \) |
| 97 | \( 1 - 5.16T + 97T^{2} \) |
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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
−8.125532314976648393406761481620, −7.45434229862280458592647244142, −6.77306473358682951415769523949, −6.31943725297044093332079680116, −5.01884180519815612788933624024, −4.36487623292247811127852186603, −3.43318682984238043942045003288, −2.59593931500891603842543126984, −1.29412400289930194479658337532, −0.26133844532667433691809998730,
0.26133844532667433691809998730, 1.29412400289930194479658337532, 2.59593931500891603842543126984, 3.43318682984238043942045003288, 4.36487623292247811127852186603, 5.01884180519815612788933624024, 6.31943725297044093332079680116, 6.77306473358682951415769523949, 7.45434229862280458592647244142, 8.125532314976648393406761481620
