| L(s) = 1 | − 1.31·5-s + 4.77·7-s − 3·9-s − 2.27·11-s − 6.09·13-s − 4.27·17-s − 19-s + 3.46·23-s − 3.27·25-s − 6.09·29-s + 2.62·31-s − 6.27·35-s + 0.837·37-s + 10.5·41-s + 10.2·43-s + 3.94·45-s + 4.77·47-s + 15.8·49-s + 10.3·53-s + 2.98·55-s + 8.54·59-s + 1.31·61-s − 14.3·63-s + 8·65-s − 8.54·67-s + 14.8·71-s − 4.27·73-s + ⋯ |
| L(s) = 1 | − 0.587·5-s + 1.80·7-s − 9-s − 0.685·11-s − 1.68·13-s − 1.03·17-s − 0.229·19-s + 0.722·23-s − 0.654·25-s − 1.13·29-s + 0.471·31-s − 1.06·35-s + 0.137·37-s + 1.64·41-s + 1.56·43-s + 0.587·45-s + 0.696·47-s + 2.26·49-s + 1.42·53-s + 0.402·55-s + 1.11·59-s + 0.168·61-s − 1.80·63-s + 0.992·65-s − 1.04·67-s + 1.75·71-s − 0.500·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368756818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368756818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 4.27T + 17T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 6.09T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 - 0.837T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 4.77T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 8.54T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 4.27T + 73T^{2} \) |
| 79 | \( 1 + 4.30T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 1.45T + 97T^{2} \) |
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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
−8.148950058443092327384742185226, −7.58467261919951614466871581755, −7.20175547870529026006492320577, −5.86741181720255623382531897124, −5.24185968637580273279790954416, −4.63157610441278403254853045631, −3.96551048948161181269159641715, −2.47716922343425848702726025959, −2.26537512162627159175577286635, −0.61493503151696599195821557926,
0.61493503151696599195821557926, 2.26537512162627159175577286635, 2.47716922343425848702726025959, 3.96551048948161181269159641715, 4.63157610441278403254853045631, 5.24185968637580273279790954416, 5.86741181720255623382531897124, 7.20175547870529026006492320577, 7.58467261919951614466871581755, 8.148950058443092327384742185226
