L(s) = 1 | − 0.654·5-s − 2.24·7-s − 3·9-s + 4.24·11-s − 4.30·13-s − 4.04·17-s + 19-s − 2.93·23-s − 4.57·25-s + 1.01·29-s + 8.52·31-s + 1.47·35-s − 8.68·37-s − 9.95·41-s + 4.61·43-s + 1.96·45-s + 9.53·47-s − 1.95·49-s + 3.71·53-s − 2.77·55-s + 0.335·59-s + 7.24·61-s + 6.73·63-s + 2.82·65-s + 0.613·67-s + 7.70·71-s + 1.41·73-s + ⋯ |
L(s) = 1 | − 0.292·5-s − 0.849·7-s − 9-s + 1.27·11-s − 1.19·13-s − 0.980·17-s + 0.229·19-s − 0.612·23-s − 0.914·25-s + 0.189·29-s + 1.53·31-s + 0.248·35-s − 1.42·37-s − 1.55·41-s + 0.704·43-s + 0.292·45-s + 1.39·47-s − 0.279·49-s + 0.510·53-s − 0.374·55-s + 0.0436·59-s + 0.927·61-s + 0.849·63-s + 0.349·65-s + 0.0749·67-s + 0.914·71-s + 0.165·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9209720648\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9209720648\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 0.654T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 + 4.30T + 13T^{2} \) |
| 17 | \( 1 + 4.04T + 17T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 - 8.52T + 31T^{2} \) |
| 37 | \( 1 + 8.68T + 37T^{2} \) |
| 41 | \( 1 + 9.95T + 41T^{2} \) |
| 43 | \( 1 - 4.61T + 43T^{2} \) |
| 47 | \( 1 - 9.53T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 - 0.335T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 0.613T + 67T^{2} \) |
| 71 | \( 1 - 7.70T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.237176387069559077007572999350, −7.19549588234557851364928849075, −6.69794966312522662210442286512, −6.05958903882889024365558942152, −5.24872415462556443298037733023, −4.33504871954382116236423492124, −3.65264613439003895154927687318, −2.81361008596563582629856639807, −1.98813996298801293998505549488, −0.47813055023363177415163328192,
0.47813055023363177415163328192, 1.98813996298801293998505549488, 2.81361008596563582629856639807, 3.65264613439003895154927687318, 4.33504871954382116236423492124, 5.24872415462556443298037733023, 6.05958903882889024365558942152, 6.69794966312522662210442286512, 7.19549588234557851364928849075, 8.237176387069559077007572999350