| L(s) = 1 | + 3·3-s − 6.81·7-s + 9·9-s − 25.7·11-s + 32.1·13-s − 9.16·17-s + 99.2·19-s − 20.4·21-s + 61.2·23-s + 27·27-s + 17.1·29-s − 74.9·31-s − 77.3·33-s − 55.9·37-s + 96.4·39-s − 54.1·41-s + 33.9·43-s + 152.·47-s − 296.·49-s − 27.5·51-s + 26.7·53-s + 297.·57-s + 567.·59-s − 249.·61-s − 61.3·63-s − 987.·67-s + 183.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.368·7-s + 0.333·9-s − 0.707·11-s + 0.686·13-s − 0.130·17-s + 1.19·19-s − 0.212·21-s + 0.555·23-s + 0.192·27-s + 0.110·29-s − 0.434·31-s − 0.408·33-s − 0.248·37-s + 0.396·39-s − 0.206·41-s + 0.120·43-s + 0.474·47-s − 0.864·49-s − 0.0755·51-s + 0.0692·53-s + 0.692·57-s + 1.25·59-s − 0.524·61-s − 0.122·63-s − 1.80·67-s + 0.320·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.646560522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646560522\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6.81T + 343T^{2} \) |
| 11 | \( 1 + 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 9.16T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 74.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 55.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 54.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 33.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 26.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 567.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 249.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 987.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 705.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 278.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 4.63T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.45e3T + 9.12e5T^{2} \) |
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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
−8.709452878678349798261631826337, −7.79449920928604197678818303488, −7.25409409214950225682775590793, −6.31753063796794124238556190054, −5.46497933671345258373039290850, −4.61954006106907230172282560498, −3.51478238614324932105633930395, −2.96418147231065894335883329923, −1.84701886886023829325987814790, −0.70301616456335483568062736676,
0.70301616456335483568062736676, 1.84701886886023829325987814790, 2.96418147231065894335883329923, 3.51478238614324932105633930395, 4.61954006106907230172282560498, 5.46497933671345258373039290850, 6.31753063796794124238556190054, 7.25409409214950225682775590793, 7.79449920928604197678818303488, 8.709452878678349798261631826337
