L(s) = 1 | − 1.48·2-s − 3.29·3-s + 0.193·4-s + 4.87·6-s + 2.22·7-s + 2.67·8-s + 7.83·9-s − 2.86·11-s − 0.638·12-s + 2.28·13-s − 3.29·14-s − 4.35·16-s − 11.5·18-s − 5.76·19-s − 7.31·21-s + 4.23·22-s + 1.58·23-s − 8.80·24-s − 3.38·26-s − 15.9·27-s + 0.430·28-s − 9.23·29-s + 1.15·31-s + 1.09·32-s + 9.41·33-s + 1.51·36-s − 0.514·37-s + ⋯ |
L(s) = 1 | − 1.04·2-s − 1.90·3-s + 0.0969·4-s + 1.99·6-s + 0.839·7-s + 0.945·8-s + 2.61·9-s − 0.862·11-s − 0.184·12-s + 0.634·13-s − 0.879·14-s − 1.08·16-s − 2.73·18-s − 1.32·19-s − 1.59·21-s + 0.903·22-s + 0.330·23-s − 1.79·24-s − 0.664·26-s − 3.06·27-s + 0.0814·28-s − 1.71·29-s + 0.207·31-s + 0.193·32-s + 1.63·33-s + 0.253·36-s − 0.0845·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3599533230\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3599533230\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 3 | \( 1 + 3.29T + 3T^{2} \) |
| 7 | \( 1 - 2.22T + 7T^{2} \) |
| 11 | \( 1 + 2.86T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 9.23T + 29T^{2} \) |
| 31 | \( 1 - 1.15T + 31T^{2} \) |
| 37 | \( 1 + 0.514T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 - 7.89T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 - 7.50T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 7.35T + 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 2.65T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 + 3.08T + 83T^{2} \) |
| 89 | \( 1 - 2.15T + 89T^{2} \) |
| 97 | \( 1 + 8.72T + 97T^{2} \) |
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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
−7.78490590359816002283138337966, −7.38884052812572137068087763052, −6.52854717263815469631110756558, −5.80555733068834065590934981387, −5.20970764628693895485687553376, −4.52090659055734811249320162064, −3.97042505257256887696242293708, −2.14867931107659164247727834803, −1.35080109743842125198703409666, −0.43968622133621833730704857131,
0.43968622133621833730704857131, 1.35080109743842125198703409666, 2.14867931107659164247727834803, 3.97042505257256887696242293708, 4.52090659055734811249320162064, 5.20970764628693895485687553376, 5.80555733068834065590934981387, 6.52854717263815469631110756558, 7.38884052812572137068087763052, 7.78490590359816002283138337966