| L(s) = 1 | − 3-s + 0.595·7-s + 9-s + 3.35·11-s − 4.76·13-s − 7.47·17-s + 6.18·19-s − 0.595·21-s + 4.40·23-s − 27-s − 2.76·29-s + 4.48·31-s − 3.35·33-s + 1.30·37-s + 4.76·39-s − 9.40·41-s − 7.59·43-s + 4.40·47-s − 6.64·49-s + 7.47·51-s − 8.20·53-s − 6.18·57-s − 2.22·59-s + 12.6·61-s + 0.595·63-s + 8.35·67-s − 4.40·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.225·7-s + 0.333·9-s + 1.01·11-s − 1.32·13-s − 1.81·17-s + 1.41·19-s − 0.130·21-s + 0.919·23-s − 0.192·27-s − 0.512·29-s + 0.806·31-s − 0.584·33-s + 0.213·37-s + 0.763·39-s − 1.46·41-s − 1.15·43-s + 0.642·47-s − 0.949·49-s + 1.04·51-s − 1.12·53-s − 0.819·57-s − 0.290·59-s + 1.61·61-s + 0.0750·63-s + 1.02·67-s − 0.530·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.595T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 4.76T + 13T^{2} \) |
| 17 | \( 1 + 7.47T + 17T^{2} \) |
| 19 | \( 1 - 6.18T + 19T^{2} \) |
| 23 | \( 1 - 4.40T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 - 4.48T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 + 9.40T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 + 8.20T + 53T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 8.35T + 67T^{2} \) |
| 71 | \( 1 - 6.79T + 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.31550968205664440607985578616, −6.82690699334135948328371593903, −6.33401224316013014768751978437, −5.16976464728162865376001929167, −4.91898118874952316074421294236, −4.07654780086677842629679207911, −3.13642214827986603913993923160, −2.16378855439398231263700152896, −1.22361955352850765718549925539, 0,
1.22361955352850765718549925539, 2.16378855439398231263700152896, 3.13642214827986603913993923160, 4.07654780086677842629679207911, 4.91898118874952316074421294236, 5.16976464728162865376001929167, 6.33401224316013014768751978437, 6.82690699334135948328371593903, 7.31550968205664440607985578616
