L(s) = 1 | + 5-s + 2·7-s + 11-s − 4·13-s + 4·17-s + 25-s + 6·29-s + 2·35-s − 2·37-s − 6·41-s − 2·43-s − 3·49-s + 10·53-s + 55-s + 12·59-s − 6·61-s − 4·65-s + 12·67-s + 16·71-s + 4·73-s + 2·77-s + 4·79-s + 2·83-s + 4·85-s − 6·89-s − 8·91-s − 2·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.301·11-s − 1.10·13-s + 0.970·17-s + 1/5·25-s + 1.11·29-s + 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s − 3/7·49-s + 1.37·53-s + 0.134·55-s + 1.56·59-s − 0.768·61-s − 0.496·65-s + 1.46·67-s + 1.89·71-s + 0.468·73-s + 0.227·77-s + 0.450·79-s + 0.219·83-s + 0.433·85-s − 0.635·89-s − 0.838·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.536193532\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.536193532\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.059472745792500999261244719781, −7.00072546750052723520756069760, −6.66471112729396818776908604070, −5.50442296888419556819148437372, −5.19310165806279374888950216968, −4.40263463844883312221176556150, −3.49404663511625023685563711414, −2.57470513606520972201761678033, −1.80381719255121776096917661031, −0.811874420240959813723434397494,
0.811874420240959813723434397494, 1.80381719255121776096917661031, 2.57470513606520972201761678033, 3.49404663511625023685563711414, 4.40263463844883312221176556150, 5.19310165806279374888950216968, 5.50442296888419556819148437372, 6.66471112729396818776908604070, 7.00072546750052723520756069760, 8.059472745792500999261244719781