| L(s) = 1 | + 1.81·2-s + 2.40·3-s + 1.27·4-s + 5-s + 4.35·6-s − 7-s − 1.30·8-s + 2.79·9-s + 1.81·10-s + 3.07·12-s + 6.34·13-s − 1.81·14-s + 2.40·15-s − 4.92·16-s + 2.07·17-s + 5.05·18-s + 3.10·19-s + 1.27·20-s − 2.40·21-s + 0.122·23-s − 3.14·24-s + 25-s + 11.4·26-s − 0.504·27-s − 1.27·28-s + 4.67·29-s + 4.35·30-s + ⋯ |
| L(s) = 1 | + 1.28·2-s + 1.38·3-s + 0.638·4-s + 0.447·5-s + 1.77·6-s − 0.377·7-s − 0.462·8-s + 0.930·9-s + 0.572·10-s + 0.887·12-s + 1.76·13-s − 0.483·14-s + 0.621·15-s − 1.23·16-s + 0.504·17-s + 1.19·18-s + 0.711·19-s + 0.285·20-s − 0.525·21-s + 0.0254·23-s − 0.642·24-s + 0.200·25-s + 2.25·26-s − 0.0970·27-s − 0.241·28-s + 0.867·29-s + 0.795·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.720194129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.720194129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.81T + 2T^{2} \) |
| 3 | \( 1 - 2.40T + 3T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 23 | \( 1 - 0.122T + 23T^{2} \) |
| 29 | \( 1 - 4.67T + 29T^{2} \) |
| 31 | \( 1 - 2.66T + 31T^{2} \) |
| 37 | \( 1 - 6.49T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 7.26T + 43T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + 4.48T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 5.24T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.451305312055823197530408063407, −7.76243079909891251362719500160, −6.65414355478647541817409792354, −6.15113868775535649797768465630, −5.37679625886571275320535515108, −4.44971465729737218520649097662, −3.58352646338488308155003363766, −3.21850671075978508705120387978, −2.44745793966874262688980567177, −1.26354397934541988482299533845,
1.26354397934541988482299533845, 2.44745793966874262688980567177, 3.21850671075978508705120387978, 3.58352646338488308155003363766, 4.44971465729737218520649097662, 5.37679625886571275320535515108, 6.15113868775535649797768465630, 6.65414355478647541817409792354, 7.76243079909891251362719500160, 8.451305312055823197530408063407
