L(s) = 1 | + 0.414·2-s − 3-s − 1.82·4-s − 5-s − 0.414·6-s + 1.41·7-s − 1.58·8-s + 9-s − 0.414·10-s + 6.24·11-s + 1.82·12-s + 0.585·13-s + 0.585·14-s + 15-s + 3·16-s + 6.82·17-s + 0.414·18-s + 1.82·20-s − 1.41·21-s + 2.58·22-s − 3.65·23-s + 1.58·24-s + 25-s + 0.242·26-s − 27-s − 2.58·28-s + 1.41·29-s + ⋯ |
L(s) = 1 | + 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.447·5-s − 0.169·6-s + 0.534·7-s − 0.560·8-s + 0.333·9-s − 0.130·10-s + 1.88·11-s + 0.527·12-s + 0.162·13-s + 0.156·14-s + 0.258·15-s + 0.750·16-s + 1.65·17-s + 0.0976·18-s + 0.408·20-s − 0.308·21-s + 0.551·22-s − 0.762·23-s + 0.323·24-s + 0.200·25-s + 0.0475·26-s − 0.192·27-s − 0.488·28-s + 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.722830398\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.722830398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 6.24T + 11T^{2} \) |
| 13 | \( 1 - 0.585T + 13T^{2} \) |
| 17 | \( 1 - 6.82T + 17T^{2} \) |
| 23 | \( 1 + 3.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 8.82T + 31T^{2} \) |
| 37 | \( 1 - 0.585T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 - 3.65T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 4.48T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 1.65T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.146882334194692924228764148594, −7.57994782542911195428636110027, −6.45922099321992712793353631351, −6.07512096931034695897690850642, −5.11376750850109856156349982156, −4.52638310859673068871195850233, −3.84098886181288094226346267997, −3.22185462012852144922909021079, −1.52098484718172889798692620556, −0.78409358858171443815299128567,
0.78409358858171443815299128567, 1.52098484718172889798692620556, 3.22185462012852144922909021079, 3.84098886181288094226346267997, 4.52638310859673068871195850233, 5.11376750850109856156349982156, 6.07512096931034695897690850642, 6.45922099321992712793353631351, 7.57994782542911195428636110027, 8.146882334194692924228764148594