L(s) = 1 | − 2.63·2-s − 3-s + 4.92·4-s + 3.86·5-s + 2.63·6-s − 7-s − 7.70·8-s + 9-s − 10.1·10-s − 3.94·11-s − 4.92·12-s + 0.666·13-s + 2.63·14-s − 3.86·15-s + 10.4·16-s − 4.28·17-s − 2.63·18-s + 0.176·19-s + 19.0·20-s + 21-s + 10.3·22-s − 5.36·23-s + 7.70·24-s + 9.94·25-s − 1.75·26-s − 27-s − 4.92·28-s + ⋯ |
L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.46·4-s + 1.72·5-s + 1.07·6-s − 0.377·7-s − 2.72·8-s + 0.333·9-s − 3.21·10-s − 1.19·11-s − 1.42·12-s + 0.184·13-s + 0.703·14-s − 0.998·15-s + 2.60·16-s − 1.03·17-s − 0.620·18-s + 0.0404·19-s + 4.26·20-s + 0.218·21-s + 2.21·22-s − 1.11·23-s + 1.57·24-s + 1.98·25-s − 0.344·26-s − 0.192·27-s − 0.931·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6095113803\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6095113803\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 13 | \( 1 - 0.666T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 - 0.176T + 19T^{2} \) |
| 23 | \( 1 + 5.36T + 23T^{2} \) |
| 29 | \( 1 + 2.72T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 37 | \( 1 - 9.69T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 3.37T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 4.21T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 - 1.67T + 73T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 + 2.95T + 83T^{2} \) |
| 89 | \( 1 - 4.67T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.74606171361924840475495481943, −7.36353835057410123525773179419, −6.46149851627029677778784345418, −5.90778617223245829989087231481, −5.64204719071216312726194760384, −4.39840418022804187995077550269, −2.87270793070850968039902776253, −2.24955195958961981762962406430, −1.66030178814629708991156140276, −0.51483249711951446412569252496,
0.51483249711951446412569252496, 1.66030178814629708991156140276, 2.24955195958961981762962406430, 2.87270793070850968039902776253, 4.39840418022804187995077550269, 5.64204719071216312726194760384, 5.90778617223245829989087231481, 6.46149851627029677778784345418, 7.36353835057410123525773179419, 7.74606171361924840475495481943