| L(s) = 1 | + 1.12·2-s − 0.740·4-s + 1.11·7-s − 3.07·8-s − 3.67·11-s − 4.05·13-s + 1.24·14-s − 1.97·16-s − 2.12·17-s − 4.06·19-s − 4.11·22-s − 6.17·23-s − 4.54·26-s − 0.824·28-s + 2.25·29-s + 10.0·31-s + 3.93·32-s − 2.38·34-s + 7.37·37-s − 4.55·38-s + 7.47·41-s + 9.24·43-s + 2.71·44-s − 6.92·46-s + 3.12·47-s − 5.75·49-s + 3.00·52-s + ⋯ |
| L(s) = 1 | + 0.793·2-s − 0.370·4-s + 0.420·7-s − 1.08·8-s − 1.10·11-s − 1.12·13-s + 0.334·14-s − 0.492·16-s − 0.514·17-s − 0.931·19-s − 0.878·22-s − 1.28·23-s − 0.891·26-s − 0.155·28-s + 0.419·29-s + 1.80·31-s + 0.696·32-s − 0.408·34-s + 1.21·37-s − 0.739·38-s + 1.16·41-s + 1.41·43-s + 0.409·44-s − 1.02·46-s + 0.456·47-s − 0.822·49-s + 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.598440820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.598440820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 + 3.67T + 11T^{2} \) |
| 13 | \( 1 + 4.05T + 13T^{2} \) |
| 17 | \( 1 + 2.12T + 17T^{2} \) |
| 19 | \( 1 + 4.06T + 19T^{2} \) |
| 23 | \( 1 + 6.17T + 23T^{2} \) |
| 29 | \( 1 - 2.25T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 - 9.24T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 - 3.50T + 53T^{2} \) |
| 59 | \( 1 - 6.59T + 59T^{2} \) |
| 61 | \( 1 + 9.10T + 61T^{2} \) |
| 67 | \( 1 - 2.62T + 67T^{2} \) |
| 71 | \( 1 + 0.660T + 71T^{2} \) |
| 73 | \( 1 - 7.47T + 73T^{2} \) |
| 79 | \( 1 - 8.53T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 13.4T + 97T^{2} \) |
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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
−8.031462623521111937806396972761, −7.58408446795862859655682955104, −6.40813319202297263430660067518, −5.94176710734505529305685368097, −5.01567379990963218002561189630, −4.55492274217876236375636077045, −3.97251343066441978936963032907, −2.66385419177469299211358629751, −2.35787613178941523516390687463, −0.56821827421959139168472260692,
0.56821827421959139168472260692, 2.35787613178941523516390687463, 2.66385419177469299211358629751, 3.97251343066441978936963032907, 4.55492274217876236375636077045, 5.01567379990963218002561189630, 5.94176710734505529305685368097, 6.40813319202297263430660067518, 7.58408446795862859655682955104, 8.031462623521111937806396972761
