| L(s) = 1 | + 1.69·2-s − 1.52·3-s + 0.863·4-s − 2.57·6-s − 1.92·8-s − 0.678·9-s − 5.95·11-s − 1.31·12-s + 6.79·13-s − 4.98·16-s − 1.14·18-s − 10.0·22-s + 2.93·24-s + 11.4·26-s + 5.60·27-s − 4.58·32-s + 9.06·33-s − 0.585·36-s − 12.1·37-s − 10.3·39-s − 5.13·44-s + 7.59·48-s − 7·49-s + 5.86·52-s + 6.04·53-s + 9.48·54-s − 15.5·61-s + ⋯ |
| L(s) = 1 | + 1.19·2-s − 0.879·3-s + 0.431·4-s − 1.05·6-s − 0.679·8-s − 0.226·9-s − 1.79·11-s − 0.379·12-s + 1.88·13-s − 1.24·16-s − 0.270·18-s − 2.14·22-s + 0.598·24-s + 2.25·26-s + 1.07·27-s − 0.810·32-s + 1.57·33-s − 0.0975·36-s − 1.99·37-s − 1.65·39-s − 0.774·44-s + 1.09·48-s − 49-s + 0.813·52-s + 0.830·53-s + 1.29·54-s − 1.99·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.470156034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.470156034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.69T + 2T^{2} \) |
| 3 | \( 1 + 1.52T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 13 | \( 1 - 6.79T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6.04T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.5T + 61T^{2} \) |
| 67 | \( 1 + 3.36T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2.99T + 97T^{2} \) |
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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
−7.57561638171826653227284017531, −6.69580397365420670601031248526, −5.98687055668754335932607898665, −5.68201344888686946042963469590, −5.02952722065185490475699282747, −4.47336923498342604978216850023, −3.40787903077264048641494715098, −3.05858237951708657298873014421, −1.89592289523569905020358650691, −0.49063262436275720731456219477,
0.49063262436275720731456219477, 1.89592289523569905020358650691, 3.05858237951708657298873014421, 3.40787903077264048641494715098, 4.47336923498342604978216850023, 5.02952722065185490475699282747, 5.68201344888686946042963469590, 5.98687055668754335932607898665, 6.69580397365420670601031248526, 7.57561638171826653227284017531
