| L(s) = 1 | + 3-s + 1.53·5-s − 0.915·7-s + 9-s + 3.09·11-s + 1.39·13-s + 1.53·15-s − 1.50·17-s − 4.21·19-s − 0.915·21-s + 4.95·23-s − 2.64·25-s + 27-s + 1.76·29-s + 3.79·31-s + 3.09·33-s − 1.40·35-s + 4.15·37-s + 1.39·39-s + 8.32·41-s + 8.45·43-s + 1.53·45-s − 6.81·47-s − 6.16·49-s − 1.50·51-s + 1.43·53-s + 4.74·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.685·5-s − 0.346·7-s + 0.333·9-s + 0.933·11-s + 0.387·13-s + 0.395·15-s − 0.363·17-s − 0.966·19-s − 0.199·21-s + 1.03·23-s − 0.529·25-s + 0.192·27-s + 0.328·29-s + 0.682·31-s + 0.539·33-s − 0.237·35-s + 0.682·37-s + 0.223·39-s + 1.30·41-s + 1.28·43-s + 0.228·45-s − 0.994·47-s − 0.880·49-s − 0.210·51-s + 0.196·53-s + 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.894802593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.894802593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 + 0.915T + 7T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 - 1.39T + 13T^{2} \) |
| 17 | \( 1 + 1.50T + 17T^{2} \) |
| 19 | \( 1 + 4.21T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 - 1.76T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 - 4.15T + 37T^{2} \) |
| 41 | \( 1 - 8.32T + 41T^{2} \) |
| 43 | \( 1 - 8.45T + 43T^{2} \) |
| 47 | \( 1 + 6.81T + 47T^{2} \) |
| 53 | \( 1 - 1.43T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 3.76T + 61T^{2} \) |
| 67 | \( 1 - 7.30T + 67T^{2} \) |
| 71 | \( 1 - 0.883T + 71T^{2} \) |
| 73 | \( 1 + 0.891T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 - 4.26T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−8.517476438757462625762327212289, −7.84583965366549109025467280728, −6.76133619734500476706478614373, −6.42706700071803908356086763439, −5.58922565703144326483323174534, −4.49538707917253446675915682392, −3.87078643208851997285045797535, −2.85521582411763115159362087928, −2.06685459868209504507168099010, −0.992704419129737678826076666887,
0.992704419129737678826076666887, 2.06685459868209504507168099010, 2.85521582411763115159362087928, 3.87078643208851997285045797535, 4.49538707917253446675915682392, 5.58922565703144326483323174534, 6.42706700071803908356086763439, 6.76133619734500476706478614373, 7.84583965366549109025467280728, 8.517476438757462625762327212289
