| L(s) = 1 | + 3-s − 3.46·5-s + 7-s + 9-s + 5.46·11-s − 3.46·13-s − 3.46·15-s + 7.46·17-s − 19-s + 21-s + 5.46·23-s + 6.99·25-s + 27-s − 6·29-s − 6.92·31-s + 5.46·33-s − 3.46·35-s + 10·37-s − 3.46·39-s − 2·41-s + 6.92·43-s − 3.46·45-s − 10.9·47-s + 49-s + 7.46·51-s + 2·53-s − 18.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.54·5-s + 0.377·7-s + 0.333·9-s + 1.64·11-s − 0.960·13-s − 0.894·15-s + 1.81·17-s − 0.229·19-s + 0.218·21-s + 1.13·23-s + 1.39·25-s + 0.192·27-s − 1.11·29-s − 1.24·31-s + 0.951·33-s − 0.585·35-s + 1.64·37-s − 0.554·39-s − 0.312·41-s + 1.05·43-s − 0.516·45-s − 1.59·47-s + 0.142·49-s + 1.04·51-s + 0.274·53-s − 2.55·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.130255522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.130255522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 23 | \( 1 - 5.46T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 + 2.92T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−7.77123754008105089067806523473, −7.56633354999644910136267475283, −6.94109368742087712520403689092, −5.93086990862838705110021157912, −4.93040176990576066936930357227, −4.26178679478939778542792197435, −3.58625787985966202761457114244, −3.07523923905604799118726174833, −1.74134081642052532664763440021, −0.76980208862776593223650390243,
0.76980208862776593223650390243, 1.74134081642052532664763440021, 3.07523923905604799118726174833, 3.58625787985966202761457114244, 4.26178679478939778542792197435, 4.93040176990576066936930357227, 5.93086990862838705110021157912, 6.94109368742087712520403689092, 7.56633354999644910136267475283, 7.77123754008105089067806523473
