| L(s) = 1 | + 2.20·2-s + 3-s + 2.87·4-s + 2.20·6-s − 7-s + 1.93·8-s + 9-s − 4.47·11-s + 2.87·12-s − 13-s − 2.20·14-s − 1.48·16-s + 2.32·17-s + 2.20·18-s − 5.93·19-s − 21-s − 9.88·22-s − 4.86·23-s + 1.93·24-s − 2.20·26-s + 27-s − 2.87·28-s − 0.932·29-s + 5.09·31-s − 7.14·32-s − 4.47·33-s + 5.13·34-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.43·4-s + 0.901·6-s − 0.377·7-s + 0.683·8-s + 0.333·9-s − 1.34·11-s + 0.830·12-s − 0.277·13-s − 0.590·14-s − 0.370·16-s + 0.563·17-s + 0.520·18-s − 1.36·19-s − 0.218·21-s − 2.10·22-s − 1.01·23-s + 0.394·24-s − 0.433·26-s + 0.192·27-s − 0.543·28-s − 0.173·29-s + 0.915·31-s − 1.26·32-s − 0.779·33-s + 0.880·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 - 2.32T + 17T^{2} \) |
| 19 | \( 1 + 5.93T + 19T^{2} \) |
| 23 | \( 1 + 4.86T + 23T^{2} \) |
| 29 | \( 1 + 0.932T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 6.50T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 0.0259T + 47T^{2} \) |
| 53 | \( 1 - 0.734T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 4.56T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 6.63T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 0.618T + 79T^{2} \) |
| 83 | \( 1 - 2.10T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 0.464T + 97T^{2} \) |
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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
−7.42160694282221124600155231509, −6.78649193718448828886515016613, −5.98920804337948892497469055528, −5.44175641089188848064060440977, −4.62752992371480739851904344564, −4.05648215274056549559247007676, −3.20897490035104827743832277196, −2.62369460062548683255117341882, −1.89905332614150511032506408120, 0,
1.89905332614150511032506408120, 2.62369460062548683255117341882, 3.20897490035104827743832277196, 4.05648215274056549559247007676, 4.62752992371480739851904344564, 5.44175641089188848064060440977, 5.98920804337948892497469055528, 6.78649193718448828886515016613, 7.42160694282221124600155231509
