| L(s) = 1 | − 2-s + 4-s + 4·5-s − 7-s − 8-s − 3·9-s − 4·10-s − 5·13-s + 14-s + 16-s − 3·17-s + 3·18-s − 19-s + 4·20-s + 9·23-s + 11·25-s + 5·26-s − 28-s + 2·29-s − 10·31-s − 32-s + 3·34-s − 4·35-s − 3·36-s − 9·37-s + 38-s − 4·40-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s − 0.377·7-s − 0.353·8-s − 9-s − 1.26·10-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.229·19-s + 0.894·20-s + 1.87·23-s + 11/5·25-s + 0.980·26-s − 0.188·28-s + 0.371·29-s − 1.79·31-s − 0.176·32-s + 0.514·34-s − 0.676·35-s − 1/2·36-s − 1.47·37-s + 0.162·38-s − 0.632·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.489800948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489800948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.765708772455712621038605306641, −7.47887638319894476711072845272, −6.85202571422676772757734776921, −6.27188701503854466251566200025, −5.32113613419338171535399937088, −5.10476339302808506059453071031, −3.45427959253860847891609558635, −2.38293153355886256664129560626, −2.20984108252189617057987884127, −0.72516821368840813854690989521,
0.72516821368840813854690989521, 2.20984108252189617057987884127, 2.38293153355886256664129560626, 3.45427959253860847891609558635, 5.10476339302808506059453071031, 5.32113613419338171535399937088, 6.27188701503854466251566200025, 6.85202571422676772757734776921, 7.47887638319894476711072845272, 8.765708772455712621038605306641
