| L(s) = 1 | + 1.36·2-s − 1.34·3-s − 0.147·4-s − 1.82·6-s − 1.74·7-s − 2.92·8-s − 1.19·9-s − 5.82·11-s + 0.198·12-s + 0.854·13-s − 2.36·14-s − 3.68·16-s − 3.37·17-s − 1.62·18-s − 7.20·19-s + 2.34·21-s − 7.92·22-s + 2.52·23-s + 3.92·24-s + 1.16·26-s + 5.63·27-s + 0.257·28-s − 7.30·29-s − 6.40·31-s + 0.834·32-s + 7.83·33-s − 4.58·34-s + ⋯ |
| L(s) = 1 | + 0.962·2-s − 0.776·3-s − 0.0738·4-s − 0.746·6-s − 0.658·7-s − 1.03·8-s − 0.397·9-s − 1.75·11-s + 0.0573·12-s + 0.236·13-s − 0.633·14-s − 0.920·16-s − 0.817·17-s − 0.382·18-s − 1.65·19-s + 0.510·21-s − 1.69·22-s + 0.526·23-s + 0.802·24-s + 0.228·26-s + 1.08·27-s + 0.0486·28-s − 1.35·29-s − 1.15·31-s + 0.147·32-s + 1.36·33-s − 0.787·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1552577780\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1552577780\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 3 | \( 1 + 1.34T + 3T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 5.82T + 11T^{2} \) |
| 13 | \( 1 - 0.854T + 13T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 19 | \( 1 + 7.20T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 + 8.32T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 - 9.91T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 8.08T + 59T^{2} \) |
| 61 | \( 1 + 4.36T + 61T^{2} \) |
| 67 | \( 1 + 3.53T + 67T^{2} \) |
| 71 | \( 1 - 7.27T + 71T^{2} \) |
| 73 | \( 1 - 4.63T + 73T^{2} \) |
| 79 | \( 1 + 1.96T + 79T^{2} \) |
| 83 | \( 1 + 0.707T + 83T^{2} \) |
| 89 | \( 1 + 7.55T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.015547207943119019440406866257, −7.15402260814112892279272007383, −6.24618376539158526204362229901, −5.89361265082365056991507858715, −5.19745919227223028192596050768, −4.62706216053512189360773969387, −3.76440958068570436186649261277, −2.92614997498842734639639313250, −2.19237292763430353616042842299, −0.17005510871581806933953770573,
0.17005510871581806933953770573, 2.19237292763430353616042842299, 2.92614997498842734639639313250, 3.76440958068570436186649261277, 4.62706216053512189360773969387, 5.19745919227223028192596050768, 5.89361265082365056991507858715, 6.24618376539158526204362229901, 7.15402260814112892279272007383, 8.015547207943119019440406866257
