| L(s) = 1 | − 0.162·2-s − 1.97·4-s + 5-s − 0.475·7-s + 0.644·8-s − 0.162·10-s + 5.19·11-s − 1.30·13-s + 0.0770·14-s + 3.84·16-s + 7.40·17-s + 6.83·19-s − 1.97·20-s − 0.842·22-s − 2.69·23-s + 25-s + 0.211·26-s + 0.938·28-s + 0.00689·29-s − 8.54·31-s − 1.91·32-s − 1.20·34-s − 0.475·35-s − 4.56·37-s − 1.10·38-s + 0.644·40-s − 3.41·41-s + ⋯ |
| L(s) = 1 | − 0.114·2-s − 0.986·4-s + 0.447·5-s − 0.179·7-s + 0.227·8-s − 0.0512·10-s + 1.56·11-s − 0.361·13-s + 0.0205·14-s + 0.960·16-s + 1.79·17-s + 1.56·19-s − 0.441·20-s − 0.179·22-s − 0.561·23-s + 0.200·25-s + 0.0414·26-s + 0.177·28-s + 0.00128·29-s − 1.53·31-s − 0.337·32-s − 0.205·34-s − 0.0803·35-s − 0.749·37-s − 0.179·38-s + 0.101·40-s − 0.532·41-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
| \(L(1)\) |
\(\approx\) |
\(1.793363889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.793363889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.162T + 2T^{2} \) |
| 7 | \( 1 + 0.475T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 - 7.40T + 17T^{2} \) |
| 19 | \( 1 - 6.83T + 19T^{2} \) |
| 23 | \( 1 + 2.69T + 23T^{2} \) |
| 29 | \( 1 - 0.00689T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 9.87T + 43T^{2} \) |
| 47 | \( 1 - 4.62T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 - 1.13T + 59T^{2} \) |
| 61 | \( 1 - 7.30T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 + 2.91T + 71T^{2} \) |
| 73 | \( 1 + 2.26T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 6.35T + 83T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.685258833351646013211737605205, −7.61016742271768827804789243786, −7.21565725695907457788414804255, −5.98628287630979239134620388049, −5.55851865590585429212043660524, −4.72376620810705676125894119643, −3.70355642119206810020935302391, −3.28145456542307878505249676004, −1.68489229668470201805370798918, −0.856155226417646382151279930522,
0.856155226417646382151279930522, 1.68489229668470201805370798918, 3.28145456542307878505249676004, 3.70355642119206810020935302391, 4.72376620810705676125894119643, 5.55851865590585429212043660524, 5.98628287630979239134620388049, 7.21565725695907457788414804255, 7.61016742271768827804789243786, 8.685258833351646013211737605205
