| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 6.42·11-s − 13-s + 14-s + 16-s − 5.42·17-s + 8.42·19-s − 6.42·22-s − 3.42·23-s − 26-s + 28-s + 3.42·29-s − 1.42·31-s + 32-s − 5.42·34-s + 8.84·37-s + 8.42·38-s − 10.4·41-s + 7·43-s − 6.42·44-s − 3.42·46-s + 10.4·47-s + 49-s − 52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.377·7-s + 0.353·8-s − 1.93·11-s − 0.277·13-s + 0.267·14-s + 0.250·16-s − 1.31·17-s + 1.93·19-s − 1.36·22-s − 0.714·23-s − 0.196·26-s + 0.188·28-s + 0.635·29-s − 0.255·31-s + 0.176·32-s − 0.930·34-s + 1.45·37-s + 1.36·38-s − 1.62·41-s + 1.06·43-s − 0.968·44-s − 0.504·46-s + 1.52·47-s + 0.142·49-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.838941140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.838941140\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 6.42T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 + 3.42T + 23T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 8.84T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 + 2.42T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 7.57T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.45894900413363941417494666452, −7.24288785383792346944843140716, −6.10389681517277424344333309978, −5.55773068564983543297440822946, −4.91399825874675983251210924876, −4.41558405753447315614809621499, −3.38190738018846224212386274471, −2.62198351390060116127385880035, −2.07790059367656303941126749460, −0.69693807153555354229597822934,
0.69693807153555354229597822934, 2.07790059367656303941126749460, 2.62198351390060116127385880035, 3.38190738018846224212386274471, 4.41558405753447315614809621499, 4.91399825874675983251210924876, 5.55773068564983543297440822946, 6.10389681517277424344333309978, 7.24288785383792346944843140716, 7.45894900413363941417494666452
