| L(s) = 1 | + 0.571·2-s − 1.67·4-s + 2.67·5-s + 2.57·7-s − 2.10·8-s + 1.52·10-s − 11-s + 4.67·13-s + 1.47·14-s + 2.14·16-s − 3.24·17-s + 3.52·19-s − 4.47·20-s − 0.571·22-s + 4.52·23-s + 2.14·25-s + 2.67·26-s − 4.30·28-s − 5.91·29-s + 4.67·31-s + 5.42·32-s − 1.85·34-s + 6.87·35-s − 11.3·37-s + 2.01·38-s − 5.61·40-s − 7.06·41-s + ⋯ |
| L(s) = 1 | + 0.404·2-s − 0.836·4-s + 1.19·5-s + 0.972·7-s − 0.742·8-s + 0.483·10-s − 0.301·11-s + 1.29·13-s + 0.393·14-s + 0.535·16-s − 0.786·17-s + 0.809·19-s − 0.999·20-s − 0.121·22-s + 0.944·23-s + 0.428·25-s + 0.524·26-s − 0.813·28-s − 1.09·29-s + 0.839·31-s + 0.959·32-s − 0.318·34-s + 1.16·35-s − 1.85·37-s + 0.327·38-s − 0.887·40-s − 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684237782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684237782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.571T + 2T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 - 2.57T + 7T^{2} \) |
| 13 | \( 1 - 4.67T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 3.52T + 19T^{2} \) |
| 23 | \( 1 - 4.52T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 - 4.67T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 7.06T + 41T^{2} \) |
| 43 | \( 1 + 2.38T + 43T^{2} \) |
| 47 | \( 1 + 9.48T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 7.96T + 59T^{2} \) |
| 61 | \( 1 - 5.16T + 61T^{2} \) |
| 67 | \( 1 - 1.61T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.75701645025776210375160220622, −10.83947391334940376397823159717, −9.811814489670179378909209917445, −8.918892819178543647955730544598, −8.185929030243038016049806149878, −6.61117425129616523122640538896, −5.49711351326388933985214188236, −4.86236356696153967914578742897, −3.40823044666790971679833213471, −1.62967540746388796897880814694,
1.62967540746388796897880814694, 3.40823044666790971679833213471, 4.86236356696153967914578742897, 5.49711351326388933985214188236, 6.61117425129616523122640538896, 8.185929030243038016049806149878, 8.918892819178543647955730544598, 9.811814489670179378909209917445, 10.83947391334940376397823159717, 11.75701645025776210375160220622
