| L(s) = 1 | + 2.08·2-s + 2.19·3-s + 2.34·4-s + 4.58·6-s − 0.992·7-s + 0.726·8-s + 1.83·9-s + 2·11-s + 5.16·12-s + 3.37·13-s − 2.06·14-s − 3.18·16-s − 2.89·17-s + 3.82·18-s − 2.58·19-s − 2.18·21-s + 4.17·22-s + 4.54·23-s + 1.59·24-s + 7.03·26-s − 2.56·27-s − 2.33·28-s − 5.38·29-s + 0.136·31-s − 8.08·32-s + 4.39·33-s − 6.03·34-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.26·3-s + 1.17·4-s + 1.87·6-s − 0.375·7-s + 0.256·8-s + 0.611·9-s + 0.603·11-s + 1.49·12-s + 0.935·13-s − 0.553·14-s − 0.795·16-s − 0.702·17-s + 0.901·18-s − 0.592·19-s − 0.476·21-s + 0.889·22-s + 0.948·23-s + 0.326·24-s + 1.37·26-s − 0.493·27-s − 0.440·28-s − 0.999·29-s + 0.0245·31-s − 1.42·32-s + 0.765·33-s − 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.361004188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.361004188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 3 | \( 1 - 2.19T + 3T^{2} \) |
| 7 | \( 1 + 0.992T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 - 4.54T + 23T^{2} \) |
| 29 | \( 1 + 5.38T + 29T^{2} \) |
| 31 | \( 1 - 0.136T + 31T^{2} \) |
| 37 | \( 1 - 2.14T + 37T^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + 4.64T + 43T^{2} \) |
| 47 | \( 1 + 9.92T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 + 2.18T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 + 0.775T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 1.77T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−10.94143123232623496408534468101, −9.437866405483968356821824184488, −8.954089238652591091399851552187, −7.967871726421320110791388198846, −6.74114971805487980228288010710, −6.08833819393060696201139696918, −4.79661576830601233838891581310, −3.78936711402077546722203334863, −3.20570169926995149903493540191, −2.05491715345655117625188202953,
2.05491715345655117625188202953, 3.20570169926995149903493540191, 3.78936711402077546722203334863, 4.79661576830601233838891581310, 6.08833819393060696201139696918, 6.74114971805487980228288010710, 7.967871726421320110791388198846, 8.954089238652591091399851552187, 9.437866405483968356821824184488, 10.94143123232623496408534468101
