| L(s) = 1 | + 3-s − 5-s − 3.27·7-s + 9-s + 5.27·11-s + 13-s − 15-s + 1.27·17-s − 2·19-s − 3.27·21-s + 7.27·23-s + 25-s + 27-s − 4·31-s + 5.27·33-s + 3.27·35-s + 0.725·37-s + 39-s + 0.725·41-s − 6.54·43-s − 45-s − 4·47-s + 3.72·49-s + 1.27·51-s + 0.725·53-s − 5.27·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.23·7-s + 0.333·9-s + 1.59·11-s + 0.277·13-s − 0.258·15-s + 0.309·17-s − 0.458·19-s − 0.714·21-s + 1.51·23-s + 0.200·25-s + 0.192·27-s − 0.718·31-s + 0.918·33-s + 0.553·35-s + 0.119·37-s + 0.160·39-s + 0.113·41-s − 0.998·43-s − 0.149·45-s − 0.583·47-s + 0.532·49-s + 0.178·51-s + 0.0995·53-s − 0.711·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.148993665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.148993665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.27T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 17 | \( 1 - 1.27T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 7.27T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 0.725T + 37T^{2} \) |
| 41 | \( 1 - 0.725T + 41T^{2} \) |
| 43 | \( 1 + 6.54T + 43T^{2} \) |
| 47 | \( 1 + 4T + 47T^{2} \) |
| 53 | \( 1 - 0.725T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 - 2.54T + 73T^{2} \) |
| 79 | \( 1 - 6.72T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 0.725T + 89T^{2} \) |
| 97 | \( 1 - 6.72T + 97T^{2} \) |
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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
−8.138403222123785471674658292413, −7.20031247086007904192065646859, −6.69099508600656563166531587058, −6.20420072796760201492824854244, −5.10441560271136886193567377622, −4.17548628624702565261622608124, −3.49542699935654225011152387765, −3.08398302976536173207143778746, −1.83225625273708586262059085227, −0.75661929101737011322710331651,
0.75661929101737011322710331651, 1.83225625273708586262059085227, 3.08398302976536173207143778746, 3.49542699935654225011152387765, 4.17548628624702565261622608124, 5.10441560271136886193567377622, 6.20420072796760201492824854244, 6.69099508600656563166531587058, 7.20031247086007904192065646859, 8.138403222123785471674658292413
