| L(s) = 1 | − 2.35·2-s + 1.56·3-s + 3.53·4-s − 3.69·6-s + 3.58·7-s − 3.60·8-s − 0.537·9-s − 2.48·11-s + 5.54·12-s + 1.25·13-s − 8.42·14-s + 1.40·16-s + 1.26·18-s − 3.63·19-s + 5.62·21-s + 5.83·22-s + 8.83·23-s − 5.65·24-s − 2.96·26-s − 5.55·27-s + 12.6·28-s + 8.75·29-s − 2.44·31-s + 3.89·32-s − 3.89·33-s − 1.89·36-s + 4.60·37-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.906·3-s + 1.76·4-s − 1.50·6-s + 1.35·7-s − 1.27·8-s − 0.179·9-s − 0.748·11-s + 1.59·12-s + 0.349·13-s − 2.25·14-s + 0.352·16-s + 0.297·18-s − 0.833·19-s + 1.22·21-s + 1.24·22-s + 1.84·23-s − 1.15·24-s − 0.580·26-s − 1.06·27-s + 2.39·28-s + 1.62·29-s − 0.438·31-s + 0.687·32-s − 0.678·33-s − 0.316·36-s + 0.756·37-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
| \(L(1)\) |
\(\approx\) |
\(1.400935185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400935185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.35T + 2T^{2} \) |
| 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 - 3.58T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 - 1.25T + 13T^{2} \) |
| 19 | \( 1 + 3.63T + 19T^{2} \) |
| 23 | \( 1 - 8.83T + 23T^{2} \) |
| 29 | \( 1 - 8.75T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 - 4.60T + 37T^{2} \) |
| 41 | \( 1 - 4.32T + 41T^{2} \) |
| 43 | \( 1 + 7.54T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 + 4.47T + 59T^{2} \) |
| 61 | \( 1 - 0.242T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 1.83T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 - 9.03T + 79T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + 2.19T + 89T^{2} \) |
| 97 | \( 1 - 9.82T + 97T^{2} \) |
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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
−8.289808003750141281263167749985, −7.57341655596206504656150552958, −6.99432024075299620117462662791, −6.06048871089613442463951499573, −5.07157912667546523582983879054, −4.36088564090166392803732655403, −3.05707055802602572247757321349, −2.46020862564156369153480937569, −1.67790255985985430068265914417, −0.75673578531378692817086055155,
0.75673578531378692817086055155, 1.67790255985985430068265914417, 2.46020862564156369153480937569, 3.05707055802602572247757321349, 4.36088564090166392803732655403, 5.07157912667546523582983879054, 6.06048871089613442463951499573, 6.99432024075299620117462662791, 7.57341655596206504656150552958, 8.289808003750141281263167749985
