| L(s) = 1 | − 2.52·2-s + 3-s + 4.37·4-s − 3.85·5-s − 2.52·6-s − 2.38·7-s − 5.99·8-s + 9-s + 9.72·10-s + 2.74·11-s + 4.37·12-s − 3.61·13-s + 6.03·14-s − 3.85·15-s + 6.38·16-s + 3.21·17-s − 2.52·18-s + 2.80·19-s − 16.8·20-s − 2.38·21-s − 6.91·22-s + 6.49·23-s − 5.99·24-s + 9.83·25-s + 9.12·26-s + 27-s − 10.4·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s + 0.577·3-s + 2.18·4-s − 1.72·5-s − 1.03·6-s − 0.902·7-s − 2.11·8-s + 0.333·9-s + 3.07·10-s + 0.826·11-s + 1.26·12-s − 1.00·13-s + 1.61·14-s − 0.994·15-s + 1.59·16-s + 0.778·17-s − 0.595·18-s + 0.643·19-s − 3.76·20-s − 0.521·21-s − 1.47·22-s + 1.35·23-s − 1.22·24-s + 1.96·25-s + 1.78·26-s + 0.192·27-s − 1.97·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5375471000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5375471000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 2.38T + 7T^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 8.36T + 29T^{2} \) |
| 31 | \( 1 - 8.59T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 8.06T + 41T^{2} \) |
| 43 | \( 1 - 7.98T + 43T^{2} \) |
| 47 | \( 1 - 0.880T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 + 0.790T + 61T^{2} \) |
| 67 | \( 1 - 1.16T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 - 9.47T + 73T^{2} \) |
| 79 | \( 1 + 3.34T + 79T^{2} \) |
| 83 | \( 1 - 6.64T + 83T^{2} \) |
| 89 | \( 1 + 6.56T + 89T^{2} \) |
| 97 | \( 1 - 3.51T + 97T^{2} \) |
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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
−7.74908987590992193710876517407, −7.52580748764799972129895380733, −6.92251985087481583452528498729, −6.27900036479876413298124071374, −4.94494787157995245660881831981, −3.98509669055175232365481946846, −3.18506900186171256430223464835, −2.72915919492245994211204176798, −1.33901649720852088325558069564, −0.50458260676635111275347043273,
0.50458260676635111275347043273, 1.33901649720852088325558069564, 2.72915919492245994211204176798, 3.18506900186171256430223464835, 3.98509669055175232365481946846, 4.94494787157995245660881831981, 6.27900036479876413298124071374, 6.92251985087481583452528498729, 7.52580748764799972129895380733, 7.74908987590992193710876517407
