| L(s) = 1 | − 2-s + 2.88·3-s + 4-s − 5-s − 2.88·6-s + 0.227·7-s − 8-s + 5.30·9-s + 10-s − 2.92·11-s + 2.88·12-s + 4.99·13-s − 0.227·14-s − 2.88·15-s + 16-s − 6.37·17-s − 5.30·18-s − 2.50·19-s − 20-s + 0.654·21-s + 2.92·22-s − 2.88·24-s + 25-s − 4.99·26-s + 6.63·27-s + 0.227·28-s − 9.29·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.66·3-s + 0.5·4-s − 0.447·5-s − 1.17·6-s + 0.0858·7-s − 0.353·8-s + 1.76·9-s + 0.316·10-s − 0.883·11-s + 0.831·12-s + 1.38·13-s − 0.0606·14-s − 0.743·15-s + 0.250·16-s − 1.54·17-s − 1.24·18-s − 0.573·19-s − 0.223·20-s + 0.142·21-s + 0.624·22-s − 0.588·24-s + 0.200·25-s − 0.980·26-s + 1.27·27-s + 0.0429·28-s − 1.72·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.88T + 3T^{2} \) |
| 7 | \( 1 - 0.227T + 7T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 + 6.37T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 29 | \( 1 + 9.29T + 29T^{2} \) |
| 31 | \( 1 + 5.42T + 31T^{2} \) |
| 37 | \( 1 + 5.22T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 + 4.33T + 43T^{2} \) |
| 47 | \( 1 - 3.62T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 9.44T + 59T^{2} \) |
| 61 | \( 1 - 4.34T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 5.65T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + 6.57T + 97T^{2} \) |
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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
−8.264133936517559502732593715368, −7.30813853129620988496004656668, −6.83458731822470234495935949711, −5.81117530656843216330206288827, −4.67466020178904787580462660598, −3.72919921225548575232737512350, −3.29007120567292328047371109663, −2.19537242251041465938805205200, −1.70088891815199627832879553636, 0,
1.70088891815199627832879553636, 2.19537242251041465938805205200, 3.29007120567292328047371109663, 3.72919921225548575232737512350, 4.67466020178904787580462660598, 5.81117530656843216330206288827, 6.83458731822470234495935949711, 7.30813853129620988496004656668, 8.264133936517559502732593715368
