| L(s) = 1 | − 2.41·2-s + 3.82·4-s + 2.82·7-s − 4.41·8-s + 0.414·11-s + 3.82·13-s − 6.82·14-s + 2.99·16-s + 0.828·17-s + 6·19-s − 0.999·22-s + 3.65·23-s − 9.24·26-s + 10.8·28-s − 29-s + 10.0·31-s + 1.58·32-s − 1.99·34-s + 4·37-s − 14.4·38-s + 4.48·41-s − 3.58·43-s + 1.58·44-s − 8.82·46-s − 3.24·47-s + 1.00·49-s + 14.6·52-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 1.06·7-s − 1.56·8-s + 0.124·11-s + 1.06·13-s − 1.82·14-s + 0.749·16-s + 0.200·17-s + 1.37·19-s − 0.213·22-s + 0.762·23-s − 1.81·26-s + 2.04·28-s − 0.185·29-s + 1.80·31-s + 0.280·32-s − 0.342·34-s + 0.657·37-s − 2.34·38-s + 0.700·41-s − 0.546·43-s + 0.239·44-s − 1.30·46-s − 0.472·47-s + 0.142·49-s + 2.03·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.308526307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.308526307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 4.48T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 + 3.24T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 - 3.65T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 8.82T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 2.41T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 4.48T + 97T^{2} \) |
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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
−8.036089293662198352514711536041, −7.67717322172192843984106078257, −6.86488413053028487341557383422, −6.17811547091710487121070194257, −5.28390884128393667741954793114, −4.44510690352983017156258280386, −3.31225715773741662367166326988, −2.40669903664075049039910749492, −1.33465613679662605040377620870, −0.914532818905671259506845220438,
0.914532818905671259506845220438, 1.33465613679662605040377620870, 2.40669903664075049039910749492, 3.31225715773741662367166326988, 4.44510690352983017156258280386, 5.28390884128393667741954793114, 6.17811547091710487121070194257, 6.86488413053028487341557383422, 7.67717322172192843984106078257, 8.036089293662198352514711536041
