| L(s) = 1 | − 3.57·3-s − 2.16·5-s − 14.1·9-s − 9.31·11-s − 56.2·13-s + 7.75·15-s − 7.42·17-s − 9.23·19-s + 129.·23-s − 120.·25-s + 147.·27-s + 9.56·29-s − 180.·31-s + 33.3·33-s + 142.·37-s + 201.·39-s − 385.·41-s + 474.·43-s + 30.7·45-s − 205.·47-s + 26.5·51-s − 558.·53-s + 20.1·55-s + 33.0·57-s − 539.·59-s + 752.·61-s + 121.·65-s + ⋯ |
| L(s) = 1 | − 0.688·3-s − 0.193·5-s − 0.525·9-s − 0.255·11-s − 1.20·13-s + 0.133·15-s − 0.105·17-s − 0.111·19-s + 1.17·23-s − 0.962·25-s + 1.05·27-s + 0.0612·29-s − 1.04·31-s + 0.175·33-s + 0.634·37-s + 0.826·39-s − 1.46·41-s + 1.68·43-s + 0.101·45-s − 0.637·47-s + 0.0729·51-s − 1.44·53-s + 0.0494·55-s + 0.0768·57-s − 1.19·59-s + 1.57·61-s + 0.232·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8455346528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8455346528\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 3.57T + 27T^{2} \) |
| 5 | \( 1 + 2.16T + 125T^{2} \) |
| 11 | \( 1 + 9.31T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 7.42T + 4.91e3T^{2} \) |
| 19 | \( 1 + 9.23T + 6.85e3T^{2} \) |
| 23 | \( 1 - 129.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 9.56T + 2.43e4T^{2} \) |
| 31 | \( 1 + 180.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 385.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 474.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 558.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 539.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 752.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 540.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 800.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 203.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 405.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 297.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 915.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.09e3T + 9.12e5T^{2} \) |
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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
−9.925504901836224048537680587050, −9.151134166627030205410779450052, −8.106634487482315085798405558447, −7.27786211539271927260092791145, −6.35573898160747509246249387317, −5.37122349253089277679232698419, −4.72278780421540686891982776148, −3.36227596865658000580234159127, −2.20159320911275726529696132285, −0.51163202735760985993350989677,
0.51163202735760985993350989677, 2.20159320911275726529696132285, 3.36227596865658000580234159127, 4.72278780421540686891982776148, 5.37122349253089277679232698419, 6.35573898160747509246249387317, 7.27786211539271927260092791145, 8.106634487482315085798405558447, 9.151134166627030205410779450052, 9.925504901836224048537680587050
