| L(s) = 1 | − 3-s − 3.41·5-s + 9-s − 4.82·11-s − 1.41·13-s + 3.41·15-s − 6.24·17-s − 1.17·19-s − 0.828·23-s + 6.65·25-s − 27-s + 8.48·29-s + 10.8·31-s + 4.82·33-s + 9.65·37-s + 1.41·39-s − 3.41·41-s + 8·43-s − 3.41·45-s + 1.17·47-s + 6.24·51-s − 9.31·53-s + 16.4·55-s + 1.17·57-s − 10.8·59-s + 5.89·61-s + 4.82·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.52·5-s + 0.333·9-s − 1.45·11-s − 0.392·13-s + 0.881·15-s − 1.51·17-s − 0.268·19-s − 0.172·23-s + 1.33·25-s − 0.192·27-s + 1.57·29-s + 1.94·31-s + 0.840·33-s + 1.58·37-s + 0.226·39-s − 0.533·41-s + 1.21·43-s − 0.508·45-s + 0.170·47-s + 0.874·51-s − 1.27·53-s + 2.22·55-s + 0.155·57-s − 1.40·59-s + 0.755·61-s + 0.598·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 11 | \( 1 + 4.82T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 1.17T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.89T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 4.82T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 - 7.31T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.45860823670195215957491632910, −6.65042461249890750529798005751, −6.17067777923531652005655270488, −5.01300652505104655717694696975, −4.59847108783631235825829125866, −4.10298621592942925142887392062, −2.93379464220138687595974545478, −2.41253881445554180388473255498, −0.816111244311408188327523533060, 0,
0.816111244311408188327523533060, 2.41253881445554180388473255498, 2.93379464220138687595974545478, 4.10298621592942925142887392062, 4.59847108783631235825829125866, 5.01300652505104655717694696975, 6.17067777923531652005655270488, 6.65042461249890750529798005751, 7.45860823670195215957491632910
