| L(s) = 1 | − 2.54·2-s + 0.596·3-s + 4.47·4-s + 4.06·5-s − 1.51·6-s − 0.751·7-s − 6.30·8-s − 2.64·9-s − 10.3·10-s + 1.59·11-s + 2.67·12-s + 0.889·13-s + 1.91·14-s + 2.42·15-s + 7.08·16-s + 6.61·17-s + 6.72·18-s + 4.89·19-s + 18.1·20-s − 0.448·21-s − 4.05·22-s + 3.07·23-s − 3.76·24-s + 11.5·25-s − 2.26·26-s − 3.36·27-s − 3.36·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 0.344·3-s + 2.23·4-s + 1.81·5-s − 0.620·6-s − 0.284·7-s − 2.22·8-s − 0.881·9-s − 3.27·10-s + 0.479·11-s + 0.771·12-s + 0.246·13-s + 0.511·14-s + 0.626·15-s + 1.77·16-s + 1.60·17-s + 1.58·18-s + 1.12·19-s + 4.06·20-s − 0.0979·21-s − 0.863·22-s + 0.640·23-s − 0.767·24-s + 2.30·25-s − 0.444·26-s − 0.648·27-s − 0.635·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494668816\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494668816\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 3 | \( 1 - 0.596T + 3T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 7 | \( 1 + 0.751T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 13 | \( 1 - 0.889T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 - 4.89T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 - 4.87T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 + 2.22T + 43T^{2} \) |
| 47 | \( 1 + 2.58T + 47T^{2} \) |
| 53 | \( 1 + 0.647T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 7.61T + 73T^{2} \) |
| 79 | \( 1 - 1.24T + 79T^{2} \) |
| 83 | \( 1 + 0.207T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 8.12T + 97T^{2} \) |
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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
−8.538115772968862326921601151197, −8.052070138307074786433263287533, −7.11327128020388616083249241131, −6.30526212055133201801863997677, −5.92807371964833660578069728791, −5.01660707751857951173326108853, −3.01644317278570824839115484220, −2.84311916386479962085271371101, −1.57387893514179856496083953164, −1.00050876487944666004493337993,
1.00050876487944666004493337993, 1.57387893514179856496083953164, 2.84311916386479962085271371101, 3.01644317278570824839115484220, 5.01660707751857951173326108853, 5.92807371964833660578069728791, 6.30526212055133201801863997677, 7.11327128020388616083249241131, 8.052070138307074786433263287533, 8.538115772968862326921601151197
