| L(s) = 1 | − 2.34·2-s − 1.75·3-s + 3.49·4-s − 1.20·5-s + 4.11·6-s + 2.58·7-s − 3.51·8-s + 0.0822·9-s + 2.83·10-s + 4.77·11-s − 6.14·12-s − 4.11·13-s − 6.06·14-s + 2.12·15-s + 1.24·16-s + 0.182·17-s − 0.192·18-s + 5.60·19-s − 4.22·20-s − 4.53·21-s − 11.2·22-s − 2.05·23-s + 6.17·24-s − 3.53·25-s + 9.65·26-s + 5.12·27-s + 9.04·28-s + ⋯ |
| L(s) = 1 | − 1.65·2-s − 1.01·3-s + 1.74·4-s − 0.540·5-s + 1.68·6-s + 0.977·7-s − 1.24·8-s + 0.0274·9-s + 0.896·10-s + 1.44·11-s − 1.77·12-s − 1.14·13-s − 1.62·14-s + 0.547·15-s + 0.312·16-s + 0.0443·17-s − 0.0454·18-s + 1.28·19-s − 0.945·20-s − 0.990·21-s − 2.38·22-s − 0.427·23-s + 1.26·24-s − 0.707·25-s + 1.89·26-s + 0.985·27-s + 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3260160254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3260160254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.34T + 2T^{2} \) |
| 3 | \( 1 + 1.75T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 4.11T + 13T^{2} \) |
| 17 | \( 1 - 0.182T + 17T^{2} \) |
| 19 | \( 1 - 5.60T + 19T^{2} \) |
| 23 | \( 1 + 2.05T + 23T^{2} \) |
| 29 | \( 1 + 8.86T + 29T^{2} \) |
| 31 | \( 1 + 6.75T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 + 6.55T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 11.7T + 53T^{2} \) |
| 59 | \( 1 - 6.93T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 - 5.47T + 67T^{2} \) |
| 71 | \( 1 - 5.38T + 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 + 7.01T + 89T^{2} \) |
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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
−7.78379978324421158446885134055, −7.03413827659220651220226469243, −6.83501579373385051364113493066, −5.63316587002441326225243834566, −5.23094145571167256630534726307, −4.24502765291572084877612500461, −3.37861052377763080755611085418, −2.01131917330891591887635670144, −1.45346434829714142650740485156, −0.39304242866221858096012735593,
0.39304242866221858096012735593, 1.45346434829714142650740485156, 2.01131917330891591887635670144, 3.37861052377763080755611085418, 4.24502765291572084877612500461, 5.23094145571167256630534726307, 5.63316587002441326225243834566, 6.83501579373385051364113493066, 7.03413827659220651220226469243, 7.78379978324421158446885134055
