| L(s) = 1 | + 2-s + 2·3-s − 4-s + 2·6-s + 7-s − 3·8-s + 9-s + 11-s − 2·12-s + 2·13-s + 14-s − 16-s + 2·17-s + 18-s + 2·21-s + 22-s + 6·23-s − 6·24-s + 2·26-s − 4·27-s − 28-s + 10·29-s + 8·31-s + 5·32-s + 2·33-s + 2·34-s − 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 0.267·14-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.436·21-s + 0.213·22-s + 1.25·23-s − 1.22·24-s + 0.392·26-s − 0.769·27-s − 0.188·28-s + 1.85·29-s + 1.43·31-s + 0.883·32-s + 0.348·33-s + 0.342·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.377632690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.377632690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.057020041464188646884032549455, −8.423103241542672438437214914469, −7.950897435934273282953942628545, −6.74937752784156403770315905179, −5.92237505604361600620899642568, −4.90218081046284059645736695951, −4.24650942233978656656073095062, −3.24049195056381263345082977707, −2.72950877701241238069066480833, −1.15226601550345924866949876011,
1.15226601550345924866949876011, 2.72950877701241238069066480833, 3.24049195056381263345082977707, 4.24650942233978656656073095062, 4.90218081046284059645736695951, 5.92237505604361600620899642568, 6.74937752784156403770315905179, 7.950897435934273282953942628545, 8.423103241542672438437214914469, 9.057020041464188646884032549455
