| L(s) = 1 | − 2.61·2-s + 4.85·4-s − 5-s + 7-s − 7.47·8-s + 2.61·10-s + 11-s − 4.23·13-s − 2.61·14-s + 9.85·16-s + 2.47·17-s − 5.47·19-s − 4.85·20-s − 2.61·22-s + 0.472·23-s − 4·25-s + 11.0·26-s + 4.85·28-s − 6.23·29-s + 8.47·31-s − 10.8·32-s − 6.47·34-s − 35-s + 3.47·37-s + 14.3·38-s + 7.47·40-s + 4.47·41-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 2.42·4-s − 0.447·5-s + 0.377·7-s − 2.64·8-s + 0.827·10-s + 0.301·11-s − 1.17·13-s − 0.699·14-s + 2.46·16-s + 0.599·17-s − 1.25·19-s − 1.08·20-s − 0.558·22-s + 0.0984·23-s − 0.800·25-s + 2.17·26-s + 0.917·28-s − 1.15·29-s + 1.52·31-s − 1.91·32-s − 1.10·34-s − 0.169·35-s + 0.570·37-s + 2.32·38-s + 1.18·40-s + 0.698·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 23 | \( 1 - 0.472T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 7.18T + 59T^{2} \) |
| 61 | \( 1 + 0.472T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.897681906301842139364814588361, −9.226799517027414843947974217601, −8.232015821263850338909008446927, −7.75028346875846517427914776097, −6.91894735363637366813466419424, −5.93100162697745718557707219480, −4.41006224078158672130249449334, −2.80893412699786579355666273064, −1.63424282427585473803587229358, 0,
1.63424282427585473803587229358, 2.80893412699786579355666273064, 4.41006224078158672130249449334, 5.93100162697745718557707219480, 6.91894735363637366813466419424, 7.75028346875846517427914776097, 8.232015821263850338909008446927, 9.226799517027414843947974217601, 9.897681906301842139364814588361
