| L(s) = 1 | − 0.159·3-s − 0.0374·5-s − 3.17·7-s − 2.97·9-s − 2.40·11-s − 6.46·13-s + 0.00595·15-s + 1.41·17-s − 3.79·19-s + 0.505·21-s − 4.99·25-s + 0.950·27-s − 1.34·29-s + 5.26·31-s + 0.382·33-s + 0.119·35-s + 2.86·37-s + 1.02·39-s + 1.53·41-s + 3.15·43-s + 0.111·45-s − 7.00·47-s + 3.10·49-s − 0.224·51-s + 12.8·53-s + 0.0901·55-s + 0.603·57-s + ⋯ |
| L(s) = 1 | − 0.0918·3-s − 0.0167·5-s − 1.20·7-s − 0.991·9-s − 0.725·11-s − 1.79·13-s + 0.00153·15-s + 0.342·17-s − 0.870·19-s + 0.110·21-s − 0.999·25-s + 0.182·27-s − 0.249·29-s + 0.945·31-s + 0.0666·33-s + 0.0201·35-s + 0.470·37-s + 0.164·39-s + 0.239·41-s + 0.480·43-s + 0.0166·45-s − 1.02·47-s + 0.443·49-s − 0.0314·51-s + 1.77·53-s + 0.0121·55-s + 0.0799·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5217294249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5217294249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.159T + 3T^{2} \) |
| 5 | \( 1 + 0.0374T + 5T^{2} \) |
| 7 | \( 1 + 3.17T + 7T^{2} \) |
| 11 | \( 1 + 2.40T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 29 | \( 1 + 1.34T + 29T^{2} \) |
| 31 | \( 1 - 5.26T + 31T^{2} \) |
| 37 | \( 1 - 2.86T + 37T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 - 3.15T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.932T + 59T^{2} \) |
| 61 | \( 1 + 4.33T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 2.28T + 97T^{2} \) |
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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
−8.293345699947136493929074451826, −7.71519111945020770497758186714, −6.89836784073841111750220686599, −6.17198091203037324817774221365, −5.50186244321519326139464372023, −4.74339863997539786131433437283, −3.72710133937433086357343763089, −2.76107060563567935955756332167, −2.31997871041112639437119635119, −0.37571011846328827779060800975,
0.37571011846328827779060800975, 2.31997871041112639437119635119, 2.76107060563567935955756332167, 3.72710133937433086357343763089, 4.74339863997539786131433437283, 5.50186244321519326139464372023, 6.17198091203037324817774221365, 6.89836784073841111750220686599, 7.71519111945020770497758186714, 8.293345699947136493929074451826
