| L(s) = 1 | + 2.61·3-s − 0.436·5-s + 3.83·9-s − 11-s − 5.35·13-s − 1.14·15-s − 7.06·17-s + 8.05·19-s − 6.66·23-s − 4.80·25-s + 2.19·27-s − 4.59·29-s + 3.04·31-s − 2.61·33-s − 9.94·37-s − 13.9·39-s − 6.37·41-s + 11.2·43-s − 1.67·45-s − 8.72·47-s − 18.4·51-s + 4.97·53-s + 0.436·55-s + 21.0·57-s − 2.27·59-s + 3.55·61-s + 2.33·65-s + ⋯ |
| L(s) = 1 | + 1.50·3-s − 0.195·5-s + 1.27·9-s − 0.301·11-s − 1.48·13-s − 0.294·15-s − 1.71·17-s + 1.84·19-s − 1.39·23-s − 0.961·25-s + 0.422·27-s − 0.853·29-s + 0.546·31-s − 0.455·33-s − 1.63·37-s − 2.24·39-s − 0.996·41-s + 1.71·43-s − 0.249·45-s − 1.27·47-s − 2.58·51-s + 0.682·53-s + 0.0588·55-s + 2.79·57-s − 0.295·59-s + 0.454·61-s + 0.289·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 5 | \( 1 + 0.436T + 5T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 - 8.05T + 19T^{2} \) |
| 23 | \( 1 + 6.66T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 + 9.94T + 37T^{2} \) |
| 41 | \( 1 + 6.37T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 8.72T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 + 2.27T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 - 9.31T + 67T^{2} \) |
| 71 | \( 1 + 0.846T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 + 8.05T + 79T^{2} \) |
| 83 | \( 1 - 9.08T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 7.77T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.050843588778324225829736286778, −7.40441556537804293062831706455, −6.96287340037362272772042047022, −5.73113892781334369520672449400, −4.86946935706880982703074051602, −4.04034348042582826378880989323, −3.30211453630956658230716027477, −2.41624645823557242595327452855, −1.87407155427191219168967405074, 0,
1.87407155427191219168967405074, 2.41624645823557242595327452855, 3.30211453630956658230716027477, 4.04034348042582826378880989323, 4.86946935706880982703074051602, 5.73113892781334369520672449400, 6.96287340037362272772042047022, 7.40441556537804293062831706455, 8.050843588778324225829736286778
