| L(s) = 1 | − 2.13·3-s − 1.27·5-s + 1.02·7-s + 1.56·9-s + 1.03·11-s + 5.61·13-s + 2.73·15-s + 3.42·17-s + 0.135·19-s − 2.18·21-s + 8.07·23-s − 3.36·25-s + 3.06·27-s − 3.88·29-s + 4.43·31-s − 2.21·33-s − 1.31·35-s − 0.834·37-s − 11.9·39-s − 7.95·41-s − 11.6·43-s − 2.00·45-s + 1.96·47-s − 5.95·49-s − 7.31·51-s + 6.11·53-s − 1.32·55-s + ⋯ |
| L(s) = 1 | − 1.23·3-s − 0.572·5-s + 0.387·7-s + 0.521·9-s + 0.313·11-s + 1.55·13-s + 0.705·15-s + 0.830·17-s + 0.0311·19-s − 0.477·21-s + 1.68·23-s − 0.672·25-s + 0.590·27-s − 0.722·29-s + 0.796·31-s − 0.386·33-s − 0.221·35-s − 0.137·37-s − 1.91·39-s − 1.24·41-s − 1.77·43-s − 0.298·45-s + 0.286·47-s − 0.850·49-s − 1.02·51-s + 0.839·53-s − 0.179·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188537638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188537638\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 2.13T + 3T^{2} \) |
| 5 | \( 1 + 1.27T + 5T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 11 | \( 1 - 1.03T + 11T^{2} \) |
| 13 | \( 1 - 5.61T + 13T^{2} \) |
| 17 | \( 1 - 3.42T + 17T^{2} \) |
| 19 | \( 1 - 0.135T + 19T^{2} \) |
| 23 | \( 1 - 8.07T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 - 4.43T + 31T^{2} \) |
| 37 | \( 1 + 0.834T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 - 1.96T + 47T^{2} \) |
| 53 | \( 1 - 6.11T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 6.84T + 71T^{2} \) |
| 73 | \( 1 - 3.52T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 1.42T + 97T^{2} \) |
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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
−8.343586840107457683113484355541, −7.75625528452436234067577469695, −6.69152354721002605986047982098, −6.34596189255615882383158567380, −5.33962253679085549054107169536, −4.95971940733812793155538684263, −3.84136059226440172881256397397, −3.23240220573368684503856069470, −1.58833358694075248969834863632, −0.71463438575689572527971651831,
0.71463438575689572527971651831, 1.58833358694075248969834863632, 3.23240220573368684503856069470, 3.84136059226440172881256397397, 4.95971940733812793155538684263, 5.33962253679085549054107169536, 6.34596189255615882383158567380, 6.69152354721002605986047982098, 7.75625528452436234067577469695, 8.343586840107457683113484355541
