| L(s) = 1 | + 3-s − 7-s + 9-s − 2·11-s − 4.47·13-s + 2.47·17-s + 2·19-s − 21-s − 4·23-s + 27-s + 0.472·29-s + 8.47·31-s − 2·33-s + 6.47·37-s − 4.47·39-s − 12.4·41-s − 6.47·43-s + 2.47·47-s + 49-s + 2.47·51-s − 2·53-s + 2·57-s − 12.4·61-s − 63-s − 10.4·67-s − 4·69-s + 3.52·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 0.333·9-s − 0.603·11-s − 1.24·13-s + 0.599·17-s + 0.458·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 0.0876·29-s + 1.52·31-s − 0.348·33-s + 1.06·37-s − 0.716·39-s − 1.94·41-s − 0.986·43-s + 0.360·47-s + 0.142·49-s + 0.346·51-s − 0.274·53-s + 0.264·57-s − 1.59·61-s − 0.125·63-s − 1.27·67-s − 0.481·69-s + 0.418·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 9.41T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.967427378184669324759081014128, −7.47780492166423502789757594684, −6.67713352177651971797603878102, −5.82516843263888697581511215778, −4.95993649845080315440281967670, −4.27534002761960482576162046688, −3.13078630377729798648968695269, −2.68141919270351332793159984935, −1.52643265782343537054209848181, 0,
1.52643265782343537054209848181, 2.68141919270351332793159984935, 3.13078630377729798648968695269, 4.27534002761960482576162046688, 4.95993649845080315440281967670, 5.82516843263888697581511215778, 6.67713352177651971797603878102, 7.47780492166423502789757594684, 7.967427378184669324759081014128
