| L(s) = 1 | + 1.31·2-s − 0.261·4-s − 5-s + 0.241·7-s − 2.98·8-s − 1.31·10-s − 1.21·11-s − 13-s + 0.318·14-s − 3.40·16-s + 6.84·17-s + 1.07·19-s + 0.261·20-s − 1.60·22-s + 5.30·23-s + 25-s − 1.31·26-s − 0.0632·28-s + 8.66·29-s + 3.90·31-s + 1.47·32-s + 9.01·34-s − 0.241·35-s + 9.38·37-s + 1.41·38-s + 2.98·40-s − 3.70·41-s + ⋯ |
| L(s) = 1 | + 0.932·2-s − 0.130·4-s − 0.447·5-s + 0.0912·7-s − 1.05·8-s − 0.416·10-s − 0.366·11-s − 0.277·13-s + 0.0851·14-s − 0.852·16-s + 1.65·17-s + 0.247·19-s + 0.0585·20-s − 0.342·22-s + 1.10·23-s + 0.200·25-s − 0.258·26-s − 0.0119·28-s + 1.60·29-s + 0.701·31-s + 0.259·32-s + 1.54·34-s − 0.0408·35-s + 1.54·37-s + 0.230·38-s + 0.471·40-s − 0.578·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.177001838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.177001838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 7 | \( 1 - 0.241T + 7T^{2} \) |
| 11 | \( 1 + 1.21T + 11T^{2} \) |
| 17 | \( 1 - 6.84T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 - 5.30T + 23T^{2} \) |
| 29 | \( 1 - 8.66T + 29T^{2} \) |
| 31 | \( 1 - 3.90T + 31T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 + 0.0769T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 + 4.02T + 53T^{2} \) |
| 59 | \( 1 - 4.40T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.00436T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 5.63T + 79T^{2} \) |
| 83 | \( 1 - 13.0T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 + 0.954T + 97T^{2} \) |
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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
−9.373838918129843562300692190011, −8.357670568991328963423184583852, −7.79144208997404189373877569475, −6.76167141948279286789330166098, −5.88083082053818020130060496068, −5.03039728802909414961287421690, −4.49614978636870397253269535505, −3.35853966917166021079712307916, −2.79742208349530528816144279259, −0.905096279643361931118964727135,
0.905096279643361931118964727135, 2.79742208349530528816144279259, 3.35853966917166021079712307916, 4.49614978636870397253269535505, 5.03039728802909414961287421690, 5.88083082053818020130060496068, 6.76167141948279286789330166098, 7.79144208997404189373877569475, 8.357670568991328963423184583852, 9.373838918129843562300692190011
