| L(s) = 1 | + 2.69·2-s + 5.27·4-s + 5-s − 1.40·7-s + 8.83·8-s + 2.69·10-s + 0.521·13-s − 3.79·14-s + 13.2·16-s + 0.132·17-s + 2·19-s + 5.27·20-s + 1.47·23-s + 25-s + 1.40·26-s − 7.42·28-s + 2.38·29-s + 4.81·31-s + 18.1·32-s + 0.357·34-s − 1.40·35-s + 1.86·37-s + 5.39·38-s + 8.83·40-s + 4.61·41-s − 1.85·43-s + 3.97·46-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s + 0.447·5-s − 0.532·7-s + 3.12·8-s + 0.852·10-s + 0.144·13-s − 1.01·14-s + 3.32·16-s + 0.0321·17-s + 0.458·19-s + 1.17·20-s + 0.307·23-s + 0.200·25-s + 0.276·26-s − 1.40·28-s + 0.442·29-s + 0.864·31-s + 3.20·32-s + 0.0613·34-s − 0.237·35-s + 0.306·37-s + 0.875·38-s + 1.39·40-s + 0.720·41-s − 0.283·43-s + 0.585·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.455694381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.455694381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.521T + 13T^{2} \) |
| 17 | \( 1 - 0.132T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 - 0.250T + 47T^{2} \) |
| 53 | \( 1 + 7.68T + 53T^{2} \) |
| 59 | \( 1 + 0.00479T + 59T^{2} \) |
| 61 | \( 1 + 4.54T + 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 2.16T + 79T^{2} \) |
| 83 | \( 1 - 4.28T + 83T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−8.154902343040787508515800155947, −7.38666604597505436487348669415, −6.50752018467517472593299214695, −6.23123223639722732186597273499, −5.33085195153697375745568897964, −4.75449860286720831301208723319, −3.88886611672412415993770973966, −3.09337926844969837190067680003, −2.46912647108449284381044597622, −1.32052206705900042066118636466,
1.32052206705900042066118636466, 2.46912647108449284381044597622, 3.09337926844969837190067680003, 3.88886611672412415993770973966, 4.75449860286720831301208723319, 5.33085195153697375745568897964, 6.23123223639722732186597273499, 6.50752018467517472593299214695, 7.38666604597505436487348669415, 8.154902343040787508515800155947
