| L(s) = 1 | − 5-s − 4.82·7-s + 2·11-s + 2.24·13-s + 4.82·17-s − 19-s + 6·23-s + 25-s − 10.4·29-s − 1.17·31-s + 4.82·35-s − 10.2·37-s + 7.65·41-s − 0.828·43-s − 0.828·47-s + 16.3·49-s + 5.07·53-s − 2·55-s − 2.34·59-s − 2.34·61-s − 2.24·65-s − 0.585·67-s − 10.8·71-s + 14.4·73-s − 9.65·77-s − 9.65·79-s − 9.31·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.82·7-s + 0.603·11-s + 0.621·13-s + 1.17·17-s − 0.229·19-s + 1.25·23-s + 0.200·25-s − 1.94·29-s − 0.210·31-s + 0.816·35-s − 1.68·37-s + 1.19·41-s − 0.126·43-s − 0.120·47-s + 2.33·49-s + 0.696·53-s − 0.269·55-s − 0.305·59-s − 0.300·61-s − 0.278·65-s − 0.0715·67-s − 1.28·71-s + 1.69·73-s − 1.10·77-s − 1.08·79-s − 1.02·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 + 0.828T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 + 9.31T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−8.291652558942667696356100301950, −7.23491622359062440134117135010, −6.89327091178823281213574633548, −5.96464623639124523534730785989, −5.40473088643548048510991073303, −4.01025103627065891807951145761, −3.54374327199999898693563382993, −2.82398481843321878060042684094, −1.30353614331230833641432697437, 0,
1.30353614331230833641432697437, 2.82398481843321878060042684094, 3.54374327199999898693563382993, 4.01025103627065891807951145761, 5.40473088643548048510991073303, 5.96464623639124523534730785989, 6.89327091178823281213574633548, 7.23491622359062440134117135010, 8.291652558942667696356100301950
