| L(s) = 1 | − 2.38·3-s − 11.6·5-s + 22.7·7-s − 21.3·9-s − 46.2·11-s − 52.2·13-s + 27.8·15-s + 17·17-s + 38.3·19-s − 54.1·21-s − 102.·23-s + 11.5·25-s + 115.·27-s − 169.·29-s + 149.·31-s + 110.·33-s − 265.·35-s − 103.·37-s + 124.·39-s + 44.4·41-s + 304.·43-s + 249.·45-s − 473.·47-s + 173.·49-s − 40.4·51-s + 223.·53-s + 540.·55-s + ⋯ |
| L(s) = 1 | − 0.458·3-s − 1.04·5-s + 1.22·7-s − 0.789·9-s − 1.26·11-s − 1.11·13-s + 0.479·15-s + 0.242·17-s + 0.462·19-s − 0.562·21-s − 0.927·23-s + 0.0923·25-s + 0.820·27-s − 1.08·29-s + 0.865·31-s + 0.580·33-s − 1.28·35-s − 0.458·37-s + 0.510·39-s + 0.169·41-s + 1.07·43-s + 0.825·45-s − 1.46·47-s + 0.507·49-s − 0.111·51-s + 0.579·53-s + 1.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6959227333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6959227333\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 2.38T + 27T^{2} \) |
| 5 | \( 1 + 11.6T + 125T^{2} \) |
| 7 | \( 1 - 22.7T + 343T^{2} \) |
| 11 | \( 1 + 46.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.2T + 2.19e3T^{2} \) |
| 19 | \( 1 - 38.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 102.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 169.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 44.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 473.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 223.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 556.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 147.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 20.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 110.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 662.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 859.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 12.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.58e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−9.539406358732130331401405040550, −8.296987137735704306697820173493, −7.894981425425519429217827447030, −7.29817804067533803299907305147, −5.87667297275603135270319689511, −5.11269319286491867226068394417, −4.47483139016385544944201887607, −3.18498021352497497767099640274, −2.08056024028459413651312391986, −0.42634533049942832991916284752,
0.42634533049942832991916284752, 2.08056024028459413651312391986, 3.18498021352497497767099640274, 4.47483139016385544944201887607, 5.11269319286491867226068394417, 5.87667297275603135270319689511, 7.29817804067533803299907305147, 7.894981425425519429217827447030, 8.296987137735704306697820173493, 9.539406358732130331401405040550
