| L(s) = 1 | + 0.317·3-s − 2.89·9-s − 3.78·11-s + 1.89·17-s + 5.97·19-s − 1.87·27-s − 1.20·33-s + 6.79·41-s + 8.48·43-s − 7·49-s + 0.603·51-s + 1.89·57-s − 14.1·59-s + 16.3·67-s − 15.6·73-s + 8.10·81-s − 17.0·83-s − 4.10·89-s − 10·97-s + 10.9·99-s − 15.0·107-s − 18.7·113-s + ⋯ |
| L(s) = 1 | + 0.183·3-s − 0.966·9-s − 1.14·11-s + 0.460·17-s + 1.37·19-s − 0.360·27-s − 0.209·33-s + 1.06·41-s + 1.29·43-s − 49-s + 0.0845·51-s + 0.251·57-s − 1.84·59-s + 1.99·67-s − 1.83·73-s + 0.900·81-s − 1.86·83-s − 0.434·89-s − 1.01·97-s + 1.10·99-s − 1.45·107-s − 1.76·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 0.317T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.78T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 5.97T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 16.3T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 4.10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| 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
−7.79130280360967418904232049454, −7.14630093955698261421195242737, −6.08316556266648971830728938643, −5.53863460502976753827261157159, −4.96494250297000239524104258498, −3.93487723900185198218735131371, −2.97072875747322522976592248663, −2.60234150203029626329223388867, −1.27324707778811480902483339128, 0,
1.27324707778811480902483339128, 2.60234150203029626329223388867, 2.97072875747322522976592248663, 3.93487723900185198218735131371, 4.96494250297000239524104258498, 5.53863460502976753827261157159, 6.08316556266648971830728938643, 7.14630093955698261421195242737, 7.79130280360967418904232049454
