| L(s) = 1 | − 3-s − 1.34·7-s + 9-s + 4.69·11-s + 2.38·13-s + 1.21·17-s + 1.03·19-s + 1.34·21-s − 6.24·23-s − 27-s − 5.59·29-s + 31-s − 4.69·33-s − 5.41·37-s − 2.38·39-s + 1.30·41-s − 5.73·43-s − 3.48·47-s − 5.18·49-s − 1.21·51-s − 2.51·53-s − 1.03·57-s + 7.86·59-s − 5.46·61-s − 1.34·63-s + 1.41·67-s + 6.24·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.509·7-s + 0.333·9-s + 1.41·11-s + 0.661·13-s + 0.294·17-s + 0.237·19-s + 0.294·21-s − 1.30·23-s − 0.192·27-s − 1.03·29-s + 0.179·31-s − 0.817·33-s − 0.890·37-s − 0.381·39-s + 0.203·41-s − 0.874·43-s − 0.508·47-s − 0.740·49-s − 0.170·51-s − 0.345·53-s − 0.137·57-s + 1.02·59-s − 0.699·61-s − 0.169·63-s + 0.173·67-s + 0.752·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.34T + 7T^{2} \) |
| 11 | \( 1 - 4.69T + 11T^{2} \) |
| 13 | \( 1 - 2.38T + 13T^{2} \) |
| 17 | \( 1 - 1.21T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 6.24T + 23T^{2} \) |
| 29 | \( 1 + 5.59T + 29T^{2} \) |
| 37 | \( 1 + 5.41T + 37T^{2} \) |
| 41 | \( 1 - 1.30T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + 2.51T + 53T^{2} \) |
| 59 | \( 1 - 7.86T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 1.81T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 + 6.69T + 97T^{2} \) |
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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
−7.18178312761559763295632633036, −6.59695453374672939918661586994, −6.06796324364477711571279439619, −5.47699086030412870890410538435, −4.52571821894659976744303498479, −3.77941638009035244273997664009, −3.31974278143259443571550461686, −1.95770045207357643489317372889, −1.22619070632607291405330109558, 0,
1.22619070632607291405330109558, 1.95770045207357643489317372889, 3.31974278143259443571550461686, 3.77941638009035244273997664009, 4.52571821894659976744303498479, 5.47699086030412870890410538435, 6.06796324364477711571279439619, 6.59695453374672939918661586994, 7.18178312761559763295632633036
