| L(s) = 1 | − 0.931·3-s + 5-s − 2.34·7-s − 2.13·9-s + 1.69·11-s + 2.14·13-s − 0.931·15-s − 1.11·17-s + 2.16·19-s + 2.18·21-s + 4.83·23-s + 25-s + 4.78·27-s − 7.77·29-s + 6.29·31-s − 1.58·33-s − 2.34·35-s − 6.49·37-s − 2.00·39-s − 10.0·41-s + 7.27·43-s − 2.13·45-s − 6.44·47-s − 1.49·49-s + 1.03·51-s + 7.11·53-s + 1.69·55-s + ⋯ |
| L(s) = 1 | − 0.538·3-s + 0.447·5-s − 0.886·7-s − 0.710·9-s + 0.511·11-s + 0.595·13-s − 0.240·15-s − 0.269·17-s + 0.497·19-s + 0.477·21-s + 1.00·23-s + 0.200·25-s + 0.920·27-s − 1.44·29-s + 1.13·31-s − 0.275·33-s − 0.396·35-s − 1.06·37-s − 0.320·39-s − 1.57·41-s + 1.10·43-s − 0.317·45-s − 0.940·47-s − 0.213·49-s + 0.144·51-s + 0.976·53-s + 0.228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.315042215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.315042215\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 3 | \( 1 + 0.931T + 3T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 - 2.14T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.16T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 - 6.29T + 31T^{2} \) |
| 37 | \( 1 + 6.49T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 + 6.44T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 0.399T + 71T^{2} \) |
| 73 | \( 1 + 6.02T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 0.442T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−8.379955521267752548825132158328, −7.30028614364273249223131130300, −6.61972057615789434170645026602, −6.10085340725007275731611038947, −5.44903062838611311492003287760, −4.69350740971252991573674021448, −3.52725114916302231239399189368, −3.02662881707344927763623471699, −1.82105859287670364766190931448, −0.63699600586213110716405101488,
0.63699600586213110716405101488, 1.82105859287670364766190931448, 3.02662881707344927763623471699, 3.52725114916302231239399189368, 4.69350740971252991573674021448, 5.44903062838611311492003287760, 6.10085340725007275731611038947, 6.61972057615789434170645026602, 7.30028614364273249223131130300, 8.379955521267752548825132158328
