| L(s) = 1 | − 2.08·2-s + 2.32·4-s − 2.29·5-s + 3.55·7-s − 0.686·8-s + 4.78·10-s + 2.81·11-s + 3.36·13-s − 7.38·14-s − 3.23·16-s − 1.06·17-s + 3.41·19-s − 5.35·20-s − 5.84·22-s + 4.88·23-s + 0.285·25-s − 6.99·26-s + 8.27·28-s + 10.1·29-s + 3.57·31-s + 8.09·32-s + 2.21·34-s − 8.16·35-s + 1.70·37-s − 7.09·38-s + 1.57·40-s − 5.33·41-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.16·4-s − 1.02·5-s + 1.34·7-s − 0.242·8-s + 1.51·10-s + 0.847·11-s + 0.932·13-s − 1.97·14-s − 0.807·16-s − 0.258·17-s + 0.782·19-s − 1.19·20-s − 1.24·22-s + 1.01·23-s + 0.0571·25-s − 1.37·26-s + 1.56·28-s + 1.88·29-s + 0.642·31-s + 1.43·32-s + 0.380·34-s − 1.38·35-s + 0.280·37-s − 1.15·38-s + 0.249·40-s − 0.833·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9387622105\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9387622105\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 5 | \( 1 + 2.29T + 5T^{2} \) |
| 7 | \( 1 - 3.55T + 7T^{2} \) |
| 11 | \( 1 - 2.81T + 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 1.06T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 - 4.88T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 3.57T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 5.33T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 7.02T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 + 9.87T + 61T^{2} \) |
| 67 | \( 1 - 1.17T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 + 3.16T + 73T^{2} \) |
| 79 | \( 1 - 7.95T + 79T^{2} \) |
| 83 | \( 1 - 9.16T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−8.822423878441648679071802862848, −8.381344625006144884298899375944, −7.83316729830175259759055197293, −7.10641985490993294882260241462, −6.30869620638867803547340794356, −4.88617940554068663865180514928, −4.28199846230099280022451707315, −3.09054965707837749502537656768, −1.59963524390048277298359079428, −0.886758517689850474466112275139,
0.886758517689850474466112275139, 1.59963524390048277298359079428, 3.09054965707837749502537656768, 4.28199846230099280022451707315, 4.88617940554068663865180514928, 6.30869620638867803547340794356, 7.10641985490993294882260241462, 7.83316729830175259759055197293, 8.381344625006144884298899375944, 8.822423878441648679071802862848
