| L(s) = 1 | − 0.713·2-s + 1.49·3-s − 1.49·4-s − 1.06·6-s + 2.49·8-s − 0.777·9-s − 3.91·11-s − 2.22·12-s + 1.91·13-s + 1.20·16-s − 3.71·17-s + 0.554·18-s + 6.69·19-s + 2.79·22-s − 1.77·23-s + 3.71·24-s − 1.36·26-s − 5.63·27-s − 7.26·29-s − 6.26·31-s − 5.84·32-s − 5.84·33-s + 2.64·34-s + 1.15·36-s − 5.06·37-s − 4.77·38-s + 2.85·39-s + ⋯ |
| L(s) = 1 | − 0.504·2-s + 0.860·3-s − 0.745·4-s − 0.434·6-s + 0.880·8-s − 0.259·9-s − 1.18·11-s − 0.641·12-s + 0.531·13-s + 0.301·16-s − 0.900·17-s + 0.130·18-s + 1.53·19-s + 0.596·22-s − 0.370·23-s + 0.758·24-s − 0.268·26-s − 1.08·27-s − 1.34·29-s − 1.12·31-s − 1.03·32-s − 1.01·33-s + 0.454·34-s + 0.193·36-s − 0.832·37-s − 0.774·38-s + 0.457·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.713T + 2T^{2} \) |
| 3 | \( 1 - 1.49T + 3T^{2} \) |
| 11 | \( 1 + 3.91T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 + 3.71T + 17T^{2} \) |
| 19 | \( 1 - 6.69T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 7.26T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 - 8.18T + 41T^{2} \) |
| 43 | \( 1 - 1.64T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 3.75T + 53T^{2} \) |
| 59 | \( 1 + 2.34T + 59T^{2} \) |
| 61 | \( 1 + 6.26T + 61T^{2} \) |
| 67 | \( 1 - 0.981T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 5.96T + 73T^{2} \) |
| 79 | \( 1 - 2.05T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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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
−9.279290485613277880164753405138, −8.554422439156943454185696032076, −7.86477131697515245707883888866, −7.30903598389159670579022143210, −5.78656504561130153250767439883, −5.07287658481676294091060549060, −3.89542307728997651931813204672, −3.03427871292206734220199167871, −1.77861670494698835573431400730, 0,
1.77861670494698835573431400730, 3.03427871292206734220199167871, 3.89542307728997651931813204672, 5.07287658481676294091060549060, 5.78656504561130153250767439883, 7.30903598389159670579022143210, 7.86477131697515245707883888866, 8.554422439156943454185696032076, 9.279290485613277880164753405138
