| L(s) = 1 | + 0.473·2-s − 1.77·4-s − 1.04·5-s + 3.83·7-s − 1.78·8-s − 0.494·10-s + 2.56·11-s + 2.32·13-s + 1.81·14-s + 2.70·16-s + 6.35·17-s + 7.47·19-s + 1.85·20-s + 1.21·22-s − 23-s − 3.90·25-s + 1.10·26-s − 6.80·28-s + 29-s + 8.99·31-s + 4.85·32-s + 3.00·34-s − 4.00·35-s − 5.17·37-s + 3.53·38-s + 1.86·40-s + 0.118·41-s + ⋯ |
| L(s) = 1 | + 0.334·2-s − 0.888·4-s − 0.467·5-s + 1.44·7-s − 0.631·8-s − 0.156·10-s + 0.773·11-s + 0.645·13-s + 0.484·14-s + 0.676·16-s + 1.54·17-s + 1.71·19-s + 0.414·20-s + 0.258·22-s − 0.208·23-s − 0.781·25-s + 0.216·26-s − 1.28·28-s + 0.185·29-s + 1.61·31-s + 0.858·32-s + 0.516·34-s − 0.676·35-s − 0.851·37-s + 0.573·38-s + 0.295·40-s + 0.0185·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524612555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524612555\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.473T + 2T^{2} \) |
| 5 | \( 1 + 1.04T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 - 6.35T + 17T^{2} \) |
| 19 | \( 1 - 7.47T + 19T^{2} \) |
| 31 | \( 1 - 8.99T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 - 0.118T + 41T^{2} \) |
| 43 | \( 1 + 2.97T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 5.29T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 - 3.51T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 + 0.286T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 + 1.13T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−8.129812750185128640514624596559, −7.66887647459978014652364567022, −6.60261195282416095954302302895, −5.67580399089777599665862827554, −5.12796327681181381343170759232, −4.51295958272353787828160754384, −3.66972010602847248836101747044, −3.17807787455389206963840797251, −1.56294120480279613182903276871, −0.908980607489805645868048871775,
0.908980607489805645868048871775, 1.56294120480279613182903276871, 3.17807787455389206963840797251, 3.66972010602847248836101747044, 4.51295958272353787828160754384, 5.12796327681181381343170759232, 5.67580399089777599665862827554, 6.60261195282416095954302302895, 7.66887647459978014652364567022, 8.129812750185128640514624596559
