L(s) = 1 | − 2-s + 3.24·3-s + 4-s + 1.66·5-s − 3.24·6-s + 4.27·7-s − 8-s + 7.51·9-s − 1.66·10-s + 1.65·11-s + 3.24·12-s + 4.89·13-s − 4.27·14-s + 5.41·15-s + 16-s − 7.55·17-s − 7.51·18-s − 0.0247·19-s + 1.66·20-s + 13.8·21-s − 1.65·22-s − 23-s − 3.24·24-s − 2.21·25-s − 4.89·26-s + 14.6·27-s + 4.27·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.87·3-s + 0.5·4-s + 0.746·5-s − 1.32·6-s + 1.61·7-s − 0.353·8-s + 2.50·9-s − 0.527·10-s + 0.497·11-s + 0.936·12-s + 1.35·13-s − 1.14·14-s + 1.39·15-s + 0.250·16-s − 1.83·17-s − 1.77·18-s − 0.00566·19-s + 0.373·20-s + 3.02·21-s − 0.352·22-s − 0.208·23-s − 0.661·24-s − 0.443·25-s − 0.960·26-s + 2.81·27-s + 0.808·28-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
\(L(1)\) |
\(\approx\) |
\(4.624714959\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.624714959\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 1.66T + 5T^{2} \) |
| 7 | \( 1 - 4.27T + 7T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 + 0.0247T + 19T^{2} \) |
| 29 | \( 1 - 8.12T + 29T^{2} \) |
| 31 | \( 1 + 9.59T + 31T^{2} \) |
| 37 | \( 1 + 0.406T + 37T^{2} \) |
| 41 | \( 1 + 7.45T + 41T^{2} \) |
| 43 | \( 1 + 5.48T + 43T^{2} \) |
| 47 | \( 1 + 2.37T + 47T^{2} \) |
| 53 | \( 1 - 0.942T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 - 0.435T + 71T^{2} \) |
| 73 | \( 1 + 7.50T + 73T^{2} \) |
| 79 | \( 1 - 4.05T + 79T^{2} \) |
| 83 | \( 1 + 6.37T + 83T^{2} \) |
| 89 | \( 1 + 6.44T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.370491492405447163934137378471, −7.66640771434218612676719791167, −6.90515641625561788349557290895, −6.25259897572183533548463816077, −5.05211998978770421818171894263, −4.22140296253458444629327328971, −3.53600337895296715418042927128, −2.41638260096747519249718736879, −1.81659558062222017159019441174, −1.38302652488148028778700700988,
1.38302652488148028778700700988, 1.81659558062222017159019441174, 2.41638260096747519249718736879, 3.53600337895296715418042927128, 4.22140296253458444629327328971, 5.05211998978770421818171894263, 6.25259897572183533548463816077, 6.90515641625561788349557290895, 7.66640771434218612676719791167, 8.370491492405447163934137378471