| L(s) = 1 | − 2.03·2-s + 2.37·3-s + 2.12·4-s − 4.81·6-s + 5.03·7-s − 0.248·8-s + 2.61·9-s − 1.90·11-s + 5.03·12-s + 1.04·13-s − 10.2·14-s − 3.74·16-s − 17-s − 5.31·18-s + 3.31·19-s + 11.9·21-s + 3.87·22-s + 0.125·23-s − 0.588·24-s − 2.11·26-s − 0.904·27-s + 10.6·28-s − 5.56·29-s − 4.99·31-s + 8.09·32-s − 4.52·33-s + 2.03·34-s + ⋯ |
| L(s) = 1 | − 1.43·2-s + 1.36·3-s + 1.06·4-s − 1.96·6-s + 1.90·7-s − 0.0877·8-s + 0.872·9-s − 0.575·11-s + 1.45·12-s + 0.288·13-s − 2.72·14-s − 0.935·16-s − 0.242·17-s − 1.25·18-s + 0.761·19-s + 2.60·21-s + 0.825·22-s + 0.0262·23-s − 0.120·24-s − 0.414·26-s − 0.174·27-s + 2.01·28-s − 1.03·29-s − 0.897·31-s + 1.43·32-s − 0.787·33-s + 0.348·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.251132834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.251132834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 2.03T + 2T^{2} \) |
| 3 | \( 1 - 2.37T + 3T^{2} \) |
| 7 | \( 1 - 5.03T + 7T^{2} \) |
| 11 | \( 1 + 1.90T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 19 | \( 1 - 3.31T + 19T^{2} \) |
| 23 | \( 1 - 0.125T + 23T^{2} \) |
| 29 | \( 1 + 5.56T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 1.56T + 37T^{2} \) |
| 41 | \( 1 - 4.72T + 41T^{2} \) |
| 43 | \( 1 - 4.46T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 - 3.28T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 1.64T + 89T^{2} \) |
| 97 | \( 1 - 7.29T + 97T^{2} \) |
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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
−10.97916727682813216458373808958, −10.04938087754672924936181200803, −8.958773011842149058780153991070, −8.639106219385790001982542634190, −7.59506586241730883142587697928, −7.51617438955552349995108202393, −5.39216983831768340094529020672, −4.10887204554274865993143797860, −2.44492992089778341553246388337, −1.49374934876020897984493821793,
1.49374934876020897984493821793, 2.44492992089778341553246388337, 4.10887204554274865993143797860, 5.39216983831768340094529020672, 7.51617438955552349995108202393, 7.59506586241730883142587697928, 8.639106219385790001982542634190, 8.958773011842149058780153991070, 10.04938087754672924936181200803, 10.97916727682813216458373808958
