| L(s) = 1 | + 1.89·3-s − 1.20·5-s − 1.24·7-s + 0.602·9-s − 11-s + 1.24·13-s − 2.29·15-s − 0.690·17-s − 19-s − 2.35·21-s + 3.95·23-s − 3.54·25-s − 4.55·27-s − 2.09·29-s + 5.19·31-s − 1.89·33-s + 1.50·35-s − 6.64·37-s + 2.35·39-s + 6.34·41-s − 10.9·43-s − 0.727·45-s − 10.2·47-s − 5.45·49-s − 1.30·51-s + 6.93·53-s + 1.20·55-s + ⋯ |
| L(s) = 1 | + 1.09·3-s − 0.540·5-s − 0.469·7-s + 0.200·9-s − 0.301·11-s + 0.344·13-s − 0.591·15-s − 0.167·17-s − 0.229·19-s − 0.514·21-s + 0.823·23-s − 0.708·25-s − 0.875·27-s − 0.388·29-s + 0.932·31-s − 0.330·33-s + 0.253·35-s − 1.09·37-s + 0.377·39-s + 0.991·41-s − 1.66·43-s − 0.108·45-s − 1.49·47-s − 0.779·49-s − 0.183·51-s + 0.953·53-s + 0.162·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.89T + 3T^{2} \) |
| 5 | \( 1 + 1.20T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + 0.690T + 17T^{2} \) |
| 23 | \( 1 - 3.95T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 - 5.19T + 31T^{2} \) |
| 37 | \( 1 + 6.64T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 6.93T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 8.96T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.81T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−8.407102111614281780235626592838, −7.65052930734526244843792797230, −6.92566002242573365318516270803, −6.08682669525073095428273439852, −5.10787065619313341514669492806, −4.12816417671618833858744102349, −3.36437545393694992129232941532, −2.77366239566713966114464564082, −1.66367824486932134097459394605, 0,
1.66367824486932134097459394605, 2.77366239566713966114464564082, 3.36437545393694992129232941532, 4.12816417671618833858744102349, 5.10787065619313341514669492806, 6.08682669525073095428273439852, 6.92566002242573365318516270803, 7.65052930734526244843792797230, 8.407102111614281780235626592838
