| L(s) = 1 | + 0.729·2-s + 1.69·3-s − 1.46·4-s + 5-s + 1.23·6-s − 7-s − 2.52·8-s − 0.132·9-s + 0.729·10-s − 2.48·12-s − 0.0510·13-s − 0.729·14-s + 1.69·15-s + 1.09·16-s + 1.59·17-s − 0.0963·18-s + 5.76·19-s − 1.46·20-s − 1.69·21-s + 8.18·23-s − 4.28·24-s + 25-s − 0.0372·26-s − 5.30·27-s + 1.46·28-s − 6.05·29-s + 1.23·30-s + ⋯ |
| L(s) = 1 | + 0.515·2-s + 0.977·3-s − 0.734·4-s + 0.447·5-s + 0.504·6-s − 0.377·7-s − 0.894·8-s − 0.0440·9-s + 0.230·10-s − 0.717·12-s − 0.0141·13-s − 0.194·14-s + 0.437·15-s + 0.272·16-s + 0.386·17-s − 0.0227·18-s + 1.32·19-s − 0.328·20-s − 0.369·21-s + 1.70·23-s − 0.874·24-s + 0.200·25-s − 0.00730·26-s − 1.02·27-s + 0.277·28-s − 1.12·29-s + 0.225·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.880501923\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.880501923\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.729T + 2T^{2} \) |
| 3 | \( 1 - 1.69T + 3T^{2} \) |
| 13 | \( 1 + 0.0510T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 5.76T + 19T^{2} \) |
| 23 | \( 1 - 8.18T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 + 8.69T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 - 6.23T + 59T^{2} \) |
| 61 | \( 1 - 4.89T + 61T^{2} \) |
| 67 | \( 1 - 0.792T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 + 9.10T + 73T^{2} \) |
| 79 | \( 1 - 2.33T + 79T^{2} \) |
| 83 | \( 1 + 7.26T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.706033106417137332449961791541, −7.57855510300585053568087186747, −7.16546747798352542692028652504, −5.74705455606298280039014933558, −5.60732636253714771179603695102, −4.58639718976093894328189405536, −3.56004490059766898382566684515, −3.19211177123568236619804424567, −2.26906705238203875151182365950, −0.859146613811287370903794380690,
0.859146613811287370903794380690, 2.26906705238203875151182365950, 3.19211177123568236619804424567, 3.56004490059766898382566684515, 4.58639718976093894328189405536, 5.60732636253714771179603695102, 5.74705455606298280039014933558, 7.16546747798352542692028652504, 7.57855510300585053568087186747, 8.706033106417137332449961791541
