| L(s) = 1 | + 2-s + 2.35·3-s + 4-s + 1.17·5-s + 2.35·6-s + 2.84·7-s + 8-s + 2.55·9-s + 1.17·10-s + 0.165·11-s + 2.35·12-s + 5.09·13-s + 2.84·14-s + 2.76·15-s + 16-s − 5.35·17-s + 2.55·18-s + 7.28·19-s + 1.17·20-s + 6.69·21-s + 0.165·22-s + 1.66·23-s + 2.35·24-s − 3.62·25-s + 5.09·26-s − 1.04·27-s + 2.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.525·5-s + 0.962·6-s + 1.07·7-s + 0.353·8-s + 0.851·9-s + 0.371·10-s + 0.0497·11-s + 0.680·12-s + 1.41·13-s + 0.759·14-s + 0.714·15-s + 0.250·16-s − 1.29·17-s + 0.602·18-s + 1.67·19-s + 0.262·20-s + 1.46·21-s + 0.0351·22-s + 0.347·23-s + 0.481·24-s − 0.724·25-s + 0.998·26-s − 0.201·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.101684952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.101684952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 2.35T + 3T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 - 2.84T + 7T^{2} \) |
| 11 | \( 1 - 0.165T + 11T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 5.35T + 17T^{2} \) |
| 19 | \( 1 - 7.28T + 19T^{2} \) |
| 23 | \( 1 - 1.66T + 23T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 37 | \( 1 - 1.74T + 37T^{2} \) |
| 41 | \( 1 + 7.51T + 41T^{2} \) |
| 43 | \( 1 + 0.422T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 7.28T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 4.50T + 89T^{2} \) |
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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
−8.237352516282573844795650299916, −7.40398905798304481183807179742, −6.74368266780407800685663304890, −5.77473495639062753118579172765, −5.19998488575921438659370720848, −4.25833436955261201528631343721, −3.61258722101686416419208898511, −2.87285728379740922999147461473, −1.95131558278572804967148617910, −1.40567766414661371069466206564,
1.40567766414661371069466206564, 1.95131558278572804967148617910, 2.87285728379740922999147461473, 3.61258722101686416419208898511, 4.25833436955261201528631343721, 5.19998488575921438659370720848, 5.77473495639062753118579172765, 6.74368266780407800685663304890, 7.40398905798304481183807179742, 8.237352516282573844795650299916
