| L(s) = 1 | − 3-s − 3.48·5-s − 3.13·7-s + 9-s + 5.14·11-s − 2.47·13-s + 3.48·15-s − 7.36·17-s + 1.17·19-s + 3.13·21-s − 4.38·23-s + 7.15·25-s − 27-s − 8.41·29-s + 4.26·31-s − 5.14·33-s + 10.9·35-s − 6.00·37-s + 2.47·39-s + 10.6·41-s − 1.62·43-s − 3.48·45-s − 8.29·47-s + 2.80·49-s + 7.36·51-s − 5.10·53-s − 17.9·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.55·5-s − 1.18·7-s + 0.333·9-s + 1.55·11-s − 0.686·13-s + 0.900·15-s − 1.78·17-s + 0.269·19-s + 0.683·21-s − 0.914·23-s + 1.43·25-s − 0.192·27-s − 1.56·29-s + 0.766·31-s − 0.896·33-s + 1.84·35-s − 0.986·37-s + 0.396·39-s + 1.66·41-s − 0.247·43-s − 0.519·45-s − 1.21·47-s + 0.401·49-s + 1.03·51-s − 0.701·53-s − 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1573070584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1573070584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 3.48T + 5T^{2} \) |
| 7 | \( 1 + 3.13T + 7T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 + 2.47T + 13T^{2} \) |
| 17 | \( 1 + 7.36T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 8.41T + 29T^{2} \) |
| 31 | \( 1 - 4.26T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 1.62T + 43T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 + 5.10T + 53T^{2} \) |
| 59 | \( 1 + 0.146T + 59T^{2} \) |
| 61 | \( 1 + 7.55T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 2.23T + 73T^{2} \) |
| 79 | \( 1 + 3.03T + 79T^{2} \) |
| 83 | \( 1 - 6.08T + 83T^{2} \) |
| 89 | \( 1 + 10.2T + 89T^{2} \) |
| 97 | \( 1 + 7.33T + 97T^{2} \) |
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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
−7.56431747306126478134933013015, −7.16940267583784774608308625666, −6.43596916022785847039225083360, −6.05577425890866598543497490630, −4.78188572396009121447689806894, −4.19813358236172134652705604375, −3.73442541171617535137646778967, −2.88319323957894395772847359959, −1.61749619600771083042013932802, −0.19970306422271198421067904064,
0.19970306422271198421067904064, 1.61749619600771083042013932802, 2.88319323957894395772847359959, 3.73442541171617535137646778967, 4.19813358236172134652705604375, 4.78188572396009121447689806894, 6.05577425890866598543497490630, 6.43596916022785847039225083360, 7.16940267583784774608308625666, 7.56431747306126478134933013015
