| L(s) = 1 | + 2.21·2-s + 3-s + 2.90·4-s + 2.21·6-s − 1.59·7-s + 1.99·8-s + 9-s + 2.90·12-s − 0.891·13-s − 3.52·14-s − 1.38·16-s − 4.43·17-s + 2.21·18-s − 5.84·19-s − 1.59·21-s − 3.27·23-s + 1.99·24-s − 1.97·26-s + 27-s − 4.62·28-s + 5.30·29-s + 8.91·31-s − 7.05·32-s − 9.82·34-s + 2.90·36-s − 5.53·37-s − 12.9·38-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.45·4-s + 0.903·6-s − 0.602·7-s + 0.705·8-s + 0.333·9-s + 0.837·12-s − 0.247·13-s − 0.942·14-s − 0.346·16-s − 1.07·17-s + 0.521·18-s − 1.34·19-s − 0.347·21-s − 0.682·23-s + 0.407·24-s − 0.387·26-s + 0.192·27-s − 0.873·28-s + 0.984·29-s + 1.60·31-s − 1.24·32-s − 1.68·34-s + 0.483·36-s − 0.909·37-s − 2.09·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 7 | \( 1 + 1.59T + 7T^{2} \) |
| 13 | \( 1 + 0.891T + 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 + 5.84T + 19T^{2} \) |
| 23 | \( 1 + 3.27T + 23T^{2} \) |
| 29 | \( 1 - 5.30T + 29T^{2} \) |
| 31 | \( 1 - 8.91T + 31T^{2} \) |
| 37 | \( 1 + 5.53T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 + 6.39T + 43T^{2} \) |
| 47 | \( 1 - 8.47T + 47T^{2} \) |
| 53 | \( 1 + 4.38T + 53T^{2} \) |
| 59 | \( 1 + 7.20T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 1.15T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−6.82571059283307450266419535269, −6.73393454273989288019018722409, −6.03259939936026231014134286659, −5.13733661425415494621846936341, −4.39136843845218333737946152453, −4.05044526086654324692009693074, −3.07865770924487841504814599299, −2.58984428303514300971147141926, −1.77222049854075451049012176230, 0,
1.77222049854075451049012176230, 2.58984428303514300971147141926, 3.07865770924487841504814599299, 4.05044526086654324692009693074, 4.39136843845218333737946152453, 5.13733661425415494621846936341, 6.03259939936026231014134286659, 6.73393454273989288019018722409, 6.82571059283307450266419535269
