L(s) = 1 | − 3-s − 7-s + 9-s − 4·11-s + 2·17-s + 2·19-s + 21-s − 23-s − 27-s + 8·31-s + 4·33-s + 6·37-s − 8·41-s + 8·43-s + 49-s − 2·51-s + 6·53-s − 2·57-s − 2·59-s − 2·61-s − 63-s + 8·67-s + 69-s + 10·71-s − 6·73-s + 4·77-s + 14·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.485·17-s + 0.458·19-s + 0.218·21-s − 0.208·23-s − 0.192·27-s + 1.43·31-s + 0.696·33-s + 0.986·37-s − 1.24·41-s + 1.21·43-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.264·57-s − 0.260·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.120·69-s + 1.18·71-s − 0.702·73-s + 0.455·77-s + 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.705104141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.705104141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.68679846940037, −13.34056089247253, −12.70086284371104, −12.37927181728946, −11.85934387130292, −11.33681778312718, −10.83647574372731, −10.31373567866242, −9.872308528110259, −9.586886825636826, −8.715050249581223, −8.289815321666461, −7.586260505631263, −7.409612317584143, −6.543060057049813, −6.172889649471300, −5.587123440128912, −5.070580454842922, −4.625445343176332, −3.879189411674468, −3.249126764132119, −2.629773873657866, −2.061789429078202, −1.026434988111728, −0.5041701605467070,
0.5041701605467070, 1.026434988111728, 2.061789429078202, 2.629773873657866, 3.249126764132119, 3.879189411674468, 4.625445343176332, 5.070580454842922, 5.587123440128912, 6.172889649471300, 6.543060057049813, 7.409612317584143, 7.586260505631263, 8.289815321666461, 8.715050249581223, 9.586886825636826, 9.872308528110259, 10.31373567866242, 10.83647574372731, 11.33681778312718, 11.85934387130292, 12.37927181728946, 12.70086284371104, 13.34056089247253, 13.68679846940037