| L(s) = 1 | − 2-s − 0.919·3-s + 4-s − 1.90·5-s + 0.919·6-s + 0.476·7-s − 8-s − 2.15·9-s + 1.90·10-s − 5.20·11-s − 0.919·12-s − 2.74·13-s − 0.476·14-s + 1.75·15-s + 16-s − 3.17·17-s + 2.15·18-s + 4.26·19-s − 1.90·20-s − 0.438·21-s + 5.20·22-s − 23-s + 0.919·24-s − 1.37·25-s + 2.74·26-s + 4.74·27-s + 0.476·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.531·3-s + 0.5·4-s − 0.851·5-s + 0.375·6-s + 0.180·7-s − 0.353·8-s − 0.717·9-s + 0.601·10-s − 1.57·11-s − 0.265·12-s − 0.760·13-s − 0.127·14-s + 0.452·15-s + 0.250·16-s − 0.770·17-s + 0.507·18-s + 0.978·19-s − 0.425·20-s − 0.0956·21-s + 1.11·22-s − 0.208·23-s + 0.187·24-s − 0.275·25-s + 0.537·26-s + 0.912·27-s + 0.0900·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008897610766\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008897610766\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 0.919T + 3T^{2} \) |
| 5 | \( 1 + 1.90T + 5T^{2} \) |
| 7 | \( 1 - 0.476T + 7T^{2} \) |
| 11 | \( 1 + 5.20T + 11T^{2} \) |
| 13 | \( 1 + 2.74T + 13T^{2} \) |
| 17 | \( 1 + 3.17T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 + 2.27T + 31T^{2} \) |
| 37 | \( 1 + 0.550T + 37T^{2} \) |
| 41 | \( 1 + 4.02T + 41T^{2} \) |
| 43 | \( 1 + 2.72T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 13.9T + 59T^{2} \) |
| 61 | \( 1 - 5.22T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.960T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 1.09T + 79T^{2} \) |
| 83 | \( 1 + 8.21T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−7.978005206500238045251350657227, −7.61626888293200086719382705085, −6.82480107111536515111158437550, −5.97401474280170850972718098126, −5.16057148799235212084960707490, −4.72345506173476453503958608483, −3.40413822574654684522386518766, −2.77654715784448764117242330382, −1.73561071892019597981915077684, −0.05363059574553886036005906068,
0.05363059574553886036005906068, 1.73561071892019597981915077684, 2.77654715784448764117242330382, 3.40413822574654684522386518766, 4.72345506173476453503958608483, 5.16057148799235212084960707490, 5.97401474280170850972718098126, 6.82480107111536515111158437550, 7.61626888293200086719382705085, 7.978005206500238045251350657227
