| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 13-s − 14-s + 16-s − 3·17-s + 2·19-s − 3·23-s + 26-s + 28-s − 3·29-s − 31-s − 32-s + 3·34-s + 2·37-s − 2·38-s + 3·41-s − 7·43-s + 3·46-s − 6·47-s + 49-s − 52-s − 9·53-s − 56-s + 3·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.458·19-s − 0.625·23-s + 0.196·26-s + 0.188·28-s − 0.557·29-s − 0.179·31-s − 0.176·32-s + 0.514·34-s + 0.328·37-s − 0.324·38-s + 0.468·41-s − 1.06·43-s + 0.442·46-s − 0.875·47-s + 1/7·49-s − 0.138·52-s − 1.23·53-s − 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.257400357303861005923811617413, −7.72552617683478337861731682443, −6.92510558510511886134207168618, −6.19467630279062369987677517167, −5.29335648956038306490960479982, −4.44000674882418286819423956557, −3.40371943274099612848009659039, −2.35705124590625994497175713453, −1.45137334552753847373459177219, 0,
1.45137334552753847373459177219, 2.35705124590625994497175713453, 3.40371943274099612848009659039, 4.44000674882418286819423956557, 5.29335648956038306490960479982, 6.19467630279062369987677517167, 6.92510558510511886134207168618, 7.72552617683478337861731682443, 8.257400357303861005923811617413
