| L(s) = 1 | − 5-s − 2·7-s − 4.24·11-s − 13-s + 4.82·17-s − 8.24·19-s − 5.07·23-s + 25-s + 9.65·29-s − 1.41·31-s + 2·35-s − 1.17·37-s + 0.828·41-s − 1.75·43-s − 2·47-s − 3·49-s + 3.17·53-s + 4.24·55-s + 5.41·59-s − 7.31·61-s + 65-s + 0.828·67-s + 9.89·71-s + 6.82·73-s + 8.48·77-s + 6.82·79-s + 2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.27·11-s − 0.277·13-s + 1.17·17-s − 1.89·19-s − 1.05·23-s + 0.200·25-s + 1.79·29-s − 0.254·31-s + 0.338·35-s − 0.192·37-s + 0.129·41-s − 0.267·43-s − 0.291·47-s − 0.428·49-s + 0.435·53-s + 0.572·55-s + 0.704·59-s − 0.936·61-s + 0.124·65-s + 0.101·67-s + 1.17·71-s + 0.799·73-s + 0.966·77-s + 0.768·79-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9097398040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9097398040\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 8.24T + 19T^{2} \) |
| 23 | \( 1 + 5.07T + 23T^{2} \) |
| 29 | \( 1 - 9.65T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 + 1.17T + 37T^{2} \) |
| 41 | \( 1 - 0.828T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 3.17T + 53T^{2} \) |
| 59 | \( 1 - 5.41T + 59T^{2} \) |
| 61 | \( 1 + 7.31T + 61T^{2} \) |
| 67 | \( 1 - 0.828T + 67T^{2} \) |
| 71 | \( 1 - 9.89T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 - 6.82T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.100141316389607925530482687267, −7.82844482018805718544550555446, −6.75469244068853948755748874586, −6.24255003490787343256139066410, −5.34199270218211662407944693852, −4.58849212238471734322637339772, −3.71431863601903252591648828882, −2.90617811780910354501540666060, −2.09060665254204218523200178541, −0.49981626230630586575389480135,
0.49981626230630586575389480135, 2.09060665254204218523200178541, 2.90617811780910354501540666060, 3.71431863601903252591648828882, 4.58849212238471734322637339772, 5.34199270218211662407944693852, 6.24255003490787343256139066410, 6.75469244068853948755748874586, 7.82844482018805718544550555446, 8.100141316389607925530482687267
