| L(s) = 1 | + 2-s − 3-s + 4-s + 1.34·5-s − 6-s − 1.25·7-s + 8-s + 9-s + 1.34·10-s + 6.18·11-s − 12-s − 13-s − 1.25·14-s − 1.34·15-s + 16-s + 0.669·17-s + 18-s + 5.57·19-s + 1.34·20-s + 1.25·21-s + 6.18·22-s − 6.51·23-s − 24-s − 3.18·25-s − 26-s − 27-s − 1.25·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.602·5-s − 0.408·6-s − 0.475·7-s + 0.353·8-s + 0.333·9-s + 0.426·10-s + 1.86·11-s − 0.288·12-s − 0.277·13-s − 0.335·14-s − 0.348·15-s + 0.250·16-s + 0.162·17-s + 0.235·18-s + 1.27·19-s + 0.301·20-s + 0.274·21-s + 1.31·22-s − 1.35·23-s − 0.204·24-s − 0.636·25-s − 0.196·26-s − 0.192·27-s − 0.237·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.507900025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.507900025\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 - 6.18T + 11T^{2} \) |
| 17 | \( 1 - 0.669T + 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 - 7.00T + 29T^{2} \) |
| 31 | \( 1 - 9.94T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 - 6.23T + 47T^{2} \) |
| 53 | \( 1 + 8.03T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 - 8.66T + 67T^{2} \) |
| 71 | \( 1 - 6.62T + 71T^{2} \) |
| 73 | \( 1 + 2.20T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 + 0.863T + 83T^{2} \) |
| 89 | \( 1 - 4.65T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.59984842275912173084355492910, −6.83252648908363872885731049270, −6.23792421902224650435309624008, −5.96244688712686492698980115098, −5.04256706988350522806529302423, −4.30658868665501211407281707880, −3.66352415838804105250531981085, −2.78706657577675678050241579780, −1.73758536384946751461568346802, −0.911048764324824881847917150978,
0.911048764324824881847917150978, 1.73758536384946751461568346802, 2.78706657577675678050241579780, 3.66352415838804105250531981085, 4.30658868665501211407281707880, 5.04256706988350522806529302423, 5.96244688712686492698980115098, 6.23792421902224650435309624008, 6.83252648908363872885731049270, 7.59984842275912173084355492910
