| L(s) = 1 | − 0.363·3-s + 5-s + 1.14·7-s − 2.86·9-s + 2.72·11-s + 4.64·13-s − 0.363·15-s − 0.858·17-s + 19-s − 0.414·21-s − 4.41·23-s + 25-s + 2.13·27-s − 9.42·29-s − 10.2·31-s − 0.990·33-s + 1.14·35-s − 6.77·37-s − 1.68·39-s − 7.55·41-s − 9.29·43-s − 2.86·45-s − 7.00·47-s − 5.69·49-s + 0.311·51-s + 8.64·53-s + 2.72·55-s + ⋯ |
| L(s) = 1 | − 0.209·3-s + 0.447·5-s + 0.431·7-s − 0.955·9-s + 0.822·11-s + 1.28·13-s − 0.0938·15-s − 0.208·17-s + 0.229·19-s − 0.0904·21-s − 0.920·23-s + 0.200·25-s + 0.410·27-s − 1.74·29-s − 1.84·31-s − 0.172·33-s + 0.192·35-s − 1.11·37-s − 0.270·39-s − 1.18·41-s − 1.41·43-s − 0.427·45-s − 1.02·47-s − 0.813·49-s + 0.0436·51-s + 1.18·53-s + 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.363T + 3T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 0.858T + 17T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.77T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 5.14T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 5.45T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.68T + 97T^{2} \) |
| 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
−7.87270129727688944758700235382, −6.76684722309831822261039085319, −6.35853596950906893728038046750, −5.45375040615467000731482052324, −5.15537733291499276353110320862, −3.69529749710956694616550375844, −3.57062490493024392806131864100, −2.06967286944176635128132144271, −1.49394789684704662201917514645, 0,
1.49394789684704662201917514645, 2.06967286944176635128132144271, 3.57062490493024392806131864100, 3.69529749710956694616550375844, 5.15537733291499276353110320862, 5.45375040615467000731482052324, 6.35853596950906893728038046750, 6.76684722309831822261039085319, 7.87270129727688944758700235382
