| L(s) = 1 | − 2-s + 1.23·3-s + 4-s + 2.90·5-s − 1.23·6-s + 4.00·7-s − 8-s − 1.48·9-s − 2.90·10-s + 6.26·11-s + 1.23·12-s − 2.27·13-s − 4.00·14-s + 3.57·15-s + 16-s − 0.819·17-s + 1.48·18-s − 7.64·19-s + 2.90·20-s + 4.92·21-s − 6.26·22-s + 2.94·23-s − 1.23·24-s + 3.44·25-s + 2.27·26-s − 5.52·27-s + 4.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.711·3-s + 0.5·4-s + 1.29·5-s − 0.502·6-s + 1.51·7-s − 0.353·8-s − 0.494·9-s − 0.918·10-s + 1.88·11-s + 0.355·12-s − 0.629·13-s − 1.06·14-s + 0.923·15-s + 0.250·16-s − 0.198·17-s + 0.349·18-s − 1.75·19-s + 0.649·20-s + 1.07·21-s − 1.33·22-s + 0.613·23-s − 0.251·24-s + 0.688·25-s + 0.445·26-s − 1.06·27-s + 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.886145311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.886145311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 11 | \( 1 - 6.26T + 11T^{2} \) |
| 13 | \( 1 + 2.27T + 13T^{2} \) |
| 17 | \( 1 + 0.819T + 17T^{2} \) |
| 19 | \( 1 + 7.64T + 19T^{2} \) |
| 23 | \( 1 - 2.94T + 23T^{2} \) |
| 29 | \( 1 - 3.43T + 29T^{2} \) |
| 31 | \( 1 - 9.79T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 + 7.67T + 41T^{2} \) |
| 43 | \( 1 + 5.80T + 43T^{2} \) |
| 47 | \( 1 - 9.83T + 47T^{2} \) |
| 53 | \( 1 + 1.60T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 3.52T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 0.390T + 71T^{2} \) |
| 73 | \( 1 - 4.15T + 73T^{2} \) |
| 79 | \( 1 - 4.69T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 4.55T + 89T^{2} \) |
| 97 | \( 1 + 5.24T + 97T^{2} \) |
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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
−8.447830352418910480321473550240, −8.191791872814564175021668244465, −6.79655004160418308720480373582, −6.58327573281277980450220121327, −5.52562448344113517165105289471, −4.72335008029123389621876048695, −3.77240730970648209046697127000, −2.42997125078115220277679622863, −2.01417948234494773609023526995, −1.14086431505898427912414528696,
1.14086431505898427912414528696, 2.01417948234494773609023526995, 2.42997125078115220277679622863, 3.77240730970648209046697127000, 4.72335008029123389621876048695, 5.52562448344113517165105289471, 6.58327573281277980450220121327, 6.79655004160418308720480373582, 8.191791872814564175021668244465, 8.447830352418910480321473550240
