| L(s) = 1 | + 1.91·2-s − 3-s + 1.67·4-s − 2.54·5-s − 1.91·6-s − 0.625·8-s + 9-s − 4.87·10-s − 5.54·11-s − 1.67·12-s + 1.83·13-s + 2.54·15-s − 4.54·16-s + 3.88·17-s + 1.91·18-s − 19-s − 4.25·20-s − 10.6·22-s + 5.54·23-s + 0.625·24-s + 1.46·25-s + 3.51·26-s − 27-s + 8.08·29-s + 4.87·30-s + 3.54·31-s − 7.46·32-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.577·3-s + 0.836·4-s − 1.13·5-s − 0.782·6-s − 0.221·8-s + 0.333·9-s − 1.54·10-s − 1.67·11-s − 0.483·12-s + 0.508·13-s + 0.656·15-s − 1.13·16-s + 0.943·17-s + 0.451·18-s − 0.229·19-s − 0.951·20-s − 2.26·22-s + 1.15·23-s + 0.127·24-s + 0.292·25-s + 0.689·26-s − 0.192·27-s + 1.50·29-s + 0.889·30-s + 0.636·31-s − 1.31·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936206294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936206294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.91T + 2T^{2} \) |
| 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 + 5.54T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 17 | \( 1 - 3.88T + 17T^{2} \) |
| 23 | \( 1 - 5.54T + 23T^{2} \) |
| 29 | \( 1 - 8.08T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 - 2.48T + 41T^{2} \) |
| 43 | \( 1 - 3.80T + 43T^{2} \) |
| 47 | \( 1 - 2.99T + 47T^{2} \) |
| 53 | \( 1 + 3.31T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 6.14T + 67T^{2} \) |
| 71 | \( 1 - 8.18T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 + 4.79T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
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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
−8.540259260913744440034710402549, −7.88321928658312667840039113867, −7.17311951144335157204688584388, −6.24300048699011067425159540064, −5.53590122558972882159917364250, −4.82176011717533220853825238608, −4.26185976626411378359440797603, −3.28121762948440251106073156930, −2.64233540573054949366710889593, −0.70722439198548647377089756241,
0.70722439198548647377089756241, 2.64233540573054949366710889593, 3.28121762948440251106073156930, 4.26185976626411378359440797603, 4.82176011717533220853825238608, 5.53590122558972882159917364250, 6.24300048699011067425159540064, 7.17311951144335157204688584388, 7.88321928658312667840039113867, 8.540259260913744440034710402549
