| L(s) = 1 | + 1.36·2-s − 0.131·4-s + 5-s + 2.67·7-s − 2.91·8-s + 1.36·10-s + 2.30·11-s − 3.08·13-s + 3.65·14-s − 3.71·16-s + 3.28·17-s + 5.71·19-s − 0.131·20-s + 3.14·22-s + 3.43·23-s + 25-s − 4.21·26-s − 0.352·28-s + 3.08·29-s − 31-s + 0.744·32-s + 4.48·34-s + 2.67·35-s + 2.62·37-s + 7.81·38-s − 2.91·40-s − 9.47·41-s + ⋯ |
| L(s) = 1 | + 0.966·2-s − 0.0659·4-s + 0.447·5-s + 1.00·7-s − 1.03·8-s + 0.432·10-s + 0.694·11-s − 0.855·13-s + 0.975·14-s − 0.929·16-s + 0.796·17-s + 1.31·19-s − 0.0294·20-s + 0.671·22-s + 0.716·23-s + 0.200·25-s − 0.827·26-s − 0.0665·28-s + 0.572·29-s − 0.179·31-s + 0.131·32-s + 0.769·34-s + 0.451·35-s + 0.432·37-s + 1.26·38-s − 0.460·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1395 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.992236258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.992236258\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.36T + 2T^{2} \) |
| 7 | \( 1 - 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 - 3.43T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 37 | \( 1 - 2.62T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 6.54T + 47T^{2} \) |
| 53 | \( 1 + 5.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 7.19T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 0.438T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 0.582T + 97T^{2} \) |
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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
−9.500943867867217937078404598277, −8.889755174330462045074451863419, −7.86565500107048762579032105545, −7.03497874837353211693396256202, −5.97405699192218912888456139285, −5.18907149636181576087924056686, −4.69883421146893580016019478495, −3.60998856903421535030756510876, −2.63086449287260746315507071188, −1.20923773989710941751895934104,
1.20923773989710941751895934104, 2.63086449287260746315507071188, 3.60998856903421535030756510876, 4.69883421146893580016019478495, 5.18907149636181576087924056686, 5.97405699192218912888456139285, 7.03497874837353211693396256202, 7.86565500107048762579032105545, 8.889755174330462045074451863419, 9.500943867867217937078404598277
