L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 4·11-s + 2·12-s − 2·13-s + 16-s + 8·17-s + 18-s − 5·19-s + 4·22-s + 23-s + 2·24-s − 2·26-s − 4·27-s + 10·29-s − 4·31-s + 32-s + 8·33-s + 8·34-s + 36-s + 37-s − 5·38-s − 4·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 1.94·17-s + 0.235·18-s − 1.14·19-s + 0.852·22-s + 0.208·23-s + 0.408·24-s − 0.392·26-s − 0.769·27-s + 1.85·29-s − 0.718·31-s + 0.176·32-s + 1.39·33-s + 1.37·34-s + 1/6·36-s + 0.164·37-s − 0.811·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.279964996\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.279964996\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−9.247832923475051677351510657525, −8.335398377062038771761517374046, −7.78890601553775332552354925353, −6.83336288190914447316190710695, −6.08448704706005188097229395161, −5.04921966442977596716512061659, −4.07433753118195086264640759969, −3.34829830899842559838586399024, −2.55382223607045619882391006072, −1.39346075938660508701472078151,
1.39346075938660508701472078151, 2.55382223607045619882391006072, 3.34829830899842559838586399024, 4.07433753118195086264640759969, 5.04921966442977596716512061659, 6.08448704706005188097229395161, 6.83336288190914447316190710695, 7.78890601553775332552354925353, 8.335398377062038771761517374046, 9.247832923475051677351510657525