| L(s) = 1 | + 2-s − 0.342·3-s + 4-s + 5-s − 0.342·6-s − 3.87·7-s + 8-s − 2.88·9-s + 10-s − 11-s − 0.342·12-s + 5.08·13-s − 3.87·14-s − 0.342·15-s + 16-s − 1.18·17-s − 2.88·18-s − 2.78·19-s + 20-s + 1.32·21-s − 22-s − 3.08·23-s − 0.342·24-s + 25-s + 5.08·26-s + 2.01·27-s − 3.87·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.198·3-s + 0.5·4-s + 0.447·5-s − 0.140·6-s − 1.46·7-s + 0.353·8-s − 0.960·9-s + 0.316·10-s − 0.301·11-s − 0.0990·12-s + 1.41·13-s − 1.03·14-s − 0.0885·15-s + 0.250·16-s − 0.286·17-s − 0.679·18-s − 0.638·19-s + 0.223·20-s + 0.289·21-s − 0.213·22-s − 0.643·23-s − 0.0700·24-s + 0.200·25-s + 0.997·26-s + 0.388·27-s − 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.279748006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.279748006\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 + 0.342T + 3T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 13 | \( 1 - 5.08T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 19 | \( 1 + 2.78T + 19T^{2} \) |
| 23 | \( 1 + 3.08T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 - 8.98T + 41T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 + 7.16T + 53T^{2} \) |
| 59 | \( 1 - 4.37T + 59T^{2} \) |
| 61 | \( 1 - 9.48T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 4.68T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 - 4.58T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.402601022530695884849723473023, −7.38318802535350955577728718883, −6.41394367701999602673767891034, −6.08700732304747804097960322086, −5.68417154581389469130108257125, −4.50696768682306178249159937774, −3.72430199516183241938630441974, −2.94741326395163947201123294724, −2.26868303840845427713906837818, −0.73242954292581245761715919022,
0.73242954292581245761715919022, 2.26868303840845427713906837818, 2.94741326395163947201123294724, 3.72430199516183241938630441974, 4.50696768682306178249159937774, 5.68417154581389469130108257125, 6.08700732304747804097960322086, 6.41394367701999602673767891034, 7.38318802535350955577728718883, 8.402601022530695884849723473023
