| L(s) = 1 | + 1.08·3-s + 0.585·5-s + 3.69·7-s − 1.82·9-s + 4.14·11-s + 3.41·13-s + 0.634·15-s + 2.82·17-s − 6.30·19-s + 4·21-s − 6.75·23-s − 4.65·25-s − 5.22·27-s + 7.41·29-s + 3.06·31-s + 4.48·33-s + 2.16·35-s + 9.07·37-s + 3.69·39-s − 4·41-s − 1.08·43-s − 1.07·45-s + 3.06·47-s + 6.65·49-s + 3.06·51-s + 4.58·53-s + 2.42·55-s + ⋯ |
| L(s) = 1 | + 0.624·3-s + 0.261·5-s + 1.39·7-s − 0.609·9-s + 1.24·11-s + 0.946·13-s + 0.163·15-s + 0.685·17-s − 1.44·19-s + 0.872·21-s − 1.40·23-s − 0.931·25-s − 1.00·27-s + 1.37·29-s + 0.549·31-s + 0.780·33-s + 0.365·35-s + 1.49·37-s + 0.591·39-s − 0.624·41-s − 0.165·43-s − 0.159·45-s + 0.446·47-s + 0.950·49-s + 0.428·51-s + 0.629·53-s + 0.327·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.455644790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.455644790\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 - 3.69T + 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 6.75T + 23T^{2} \) |
| 29 | \( 1 - 7.41T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 9.07T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 1.08T + 43T^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 - 4.58T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 + 1.07T + 61T^{2} \) |
| 67 | \( 1 - 1.97T + 67T^{2} \) |
| 71 | \( 1 + 8.02T + 71T^{2} \) |
| 73 | \( 1 + 6.48T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 - 4.82T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−9.901737894378749545138942968076, −8.853896894844756469920230056290, −8.355530115495223313565532289630, −7.79380577458599340032547642662, −6.38157768700889576918052761335, −5.82120843789372009944298462339, −4.47773709437678088260033584226, −3.78169473542701537862071289041, −2.37617883816608787264309684634, −1.39136315239127214852981826771,
1.39136315239127214852981826771, 2.37617883816608787264309684634, 3.78169473542701537862071289041, 4.47773709437678088260033584226, 5.82120843789372009944298462339, 6.38157768700889576918052761335, 7.79380577458599340032547642662, 8.355530115495223313565532289630, 8.853896894844756469920230056290, 9.901737894378749545138942968076
