| L(s) = 1 | + 2-s + 3-s + 4-s + 1.38·5-s + 6-s − 3.20·7-s + 8-s + 9-s + 1.38·10-s + 5.82·11-s + 12-s − 0.957·13-s − 3.20·14-s + 1.38·15-s + 16-s − 3.20·17-s + 18-s + 3.20·19-s + 1.38·20-s − 3.20·21-s + 5.82·22-s − 23-s + 24-s − 3.08·25-s − 0.957·26-s + 27-s − 3.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.619·5-s + 0.408·6-s − 1.21·7-s + 0.353·8-s + 0.333·9-s + 0.437·10-s + 1.75·11-s + 0.288·12-s − 0.265·13-s − 0.856·14-s + 0.357·15-s + 0.250·16-s − 0.777·17-s + 0.235·18-s + 0.735·19-s + 0.309·20-s − 0.699·21-s + 1.24·22-s − 0.208·23-s + 0.204·24-s − 0.616·25-s − 0.187·26-s + 0.192·27-s − 0.605·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.208465572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.208465572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 + 0.957T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 3.20T + 19T^{2} \) |
| 31 | \( 1 - 5.72T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 + 0.0838T + 41T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 - 8.60T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8.39T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 9.74T + 73T^{2} \) |
| 79 | \( 1 + 8.33T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 8.98T + 89T^{2} \) |
| 97 | \( 1 - 7.71T + 97T^{2} \) |
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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
−8.607250613063000775493068800738, −7.46228480782137882280889406906, −6.83428656669093386248928786111, −6.21350795132052184310064649521, −5.68078310836492675432732822605, −4.35569426831636427665935057582, −3.93599366292903357364614241732, −2.97527158551406448730150668195, −2.26911511414332033860136081135, −1.08785004485030173178600087229,
1.08785004485030173178600087229, 2.26911511414332033860136081135, 2.97527158551406448730150668195, 3.93599366292903357364614241732, 4.35569426831636427665935057582, 5.68078310836492675432732822605, 6.21350795132052184310064649521, 6.83428656669093386248928786111, 7.46228480782137882280889406906, 8.607250613063000775493068800738
