| L(s) = 1 | − 0.231·2-s − 1.94·4-s − 3.70·5-s − 0.911·7-s + 0.912·8-s + 0.857·10-s + 11-s − 1.28·13-s + 0.210·14-s + 3.68·16-s + 7.36·17-s + 5.91·19-s + 7.21·20-s − 0.231·22-s − 6.40·23-s + 8.74·25-s + 0.298·26-s + 1.77·28-s − 9.58·29-s − 8.65·31-s − 2.67·32-s − 1.70·34-s + 3.37·35-s − 8.74·37-s − 1.36·38-s − 3.38·40-s − 6.54·41-s + ⋯ |
| L(s) = 1 | − 0.163·2-s − 0.973·4-s − 1.65·5-s − 0.344·7-s + 0.322·8-s + 0.271·10-s + 0.301·11-s − 0.357·13-s + 0.0563·14-s + 0.920·16-s + 1.78·17-s + 1.35·19-s + 1.61·20-s − 0.0493·22-s − 1.33·23-s + 1.74·25-s + 0.0585·26-s + 0.335·28-s − 1.77·29-s − 1.55·31-s − 0.473·32-s − 0.292·34-s + 0.570·35-s − 1.43·37-s − 0.221·38-s − 0.534·40-s − 1.02·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4972333754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4972333754\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.231T + 2T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 0.911T + 7T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 7.36T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 + 6.40T + 23T^{2} \) |
| 29 | \( 1 + 9.58T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 - 9.59T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 5.01T + 53T^{2} \) |
| 59 | \( 1 + 1.55T + 59T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 - 0.506T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 + 5.28T + 79T^{2} \) |
| 83 | \( 1 + 7.43T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−7.924596374062747487185167584371, −7.58866600601016836375102969611, −7.01924054918251773804005510841, −5.60750972412211380648423368213, −5.30377027750610365184036307740, −4.22992755434547474909443997993, −3.53722078016262997921803953382, −3.37125103941500529275129872942, −1.56156717444436071913658005399, −0.39700894289407426371349317147,
0.39700894289407426371349317147, 1.56156717444436071913658005399, 3.37125103941500529275129872942, 3.53722078016262997921803953382, 4.22992755434547474909443997993, 5.30377027750610365184036307740, 5.60750972412211380648423368213, 7.01924054918251773804005510841, 7.58866600601016836375102969611, 7.924596374062747487185167584371
