| L(s) = 1 | + 1.38·2-s − 1.77·3-s − 0.0920·4-s − 2.44·6-s − 7-s − 2.88·8-s + 0.141·9-s − 0.183·11-s + 0.163·12-s − 1.86·13-s − 1.38·14-s − 3.80·16-s + 0.498·17-s + 0.194·18-s − 1.56·19-s + 1.77·21-s − 0.253·22-s − 23-s + 5.12·24-s − 2.57·26-s + 5.06·27-s + 0.0920·28-s − 7.91·29-s − 1.77·31-s + 0.520·32-s + 0.324·33-s + 0.688·34-s + ⋯ |
| L(s) = 1 | + 0.976·2-s − 1.02·3-s − 0.0460·4-s − 0.999·6-s − 0.377·7-s − 1.02·8-s + 0.0470·9-s − 0.0552·11-s + 0.0471·12-s − 0.516·13-s − 0.369·14-s − 0.951·16-s + 0.120·17-s + 0.0459·18-s − 0.358·19-s + 0.386·21-s − 0.0539·22-s − 0.208·23-s + 1.04·24-s − 0.504·26-s + 0.975·27-s + 0.0173·28-s − 1.46·29-s − 0.318·31-s + 0.0919·32-s + 0.0565·33-s + 0.118·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9239779456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9239779456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.38T + 2T^{2} \) |
| 3 | \( 1 + 1.77T + 3T^{2} \) |
| 11 | \( 1 + 0.183T + 11T^{2} \) |
| 13 | \( 1 + 1.86T + 13T^{2} \) |
| 17 | \( 1 - 0.498T + 17T^{2} \) |
| 19 | \( 1 + 1.56T + 19T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 - 7.58T + 37T^{2} \) |
| 41 | \( 1 + 8.71T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 + 5.02T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 5.67T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 9.94T + 79T^{2} \) |
| 83 | \( 1 - 5.39T + 83T^{2} \) |
| 89 | \( 1 - 7.63T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.411333323778003219163852395825, −7.55114765024514048105465986575, −6.47644092059556997204076465019, −6.20492406449120148229437456821, −5.24443773926472598382900326524, −4.97221567045094210253044057111, −3.95258969387679386951720835466, −3.24137194161892170416899966054, −2.16569104327429599037691876937, −0.47575255760236360184073508667,
0.47575255760236360184073508667, 2.16569104327429599037691876937, 3.24137194161892170416899966054, 3.95258969387679386951720835466, 4.97221567045094210253044057111, 5.24443773926472598382900326524, 6.20492406449120148229437456821, 6.47644092059556997204076465019, 7.55114765024514048105465986575, 8.411333323778003219163852395825
