| L(s) = 1 | + 1.12·2-s − 2.15·3-s − 0.730·4-s + 5-s − 2.42·6-s − 4.58·7-s − 3.07·8-s + 1.63·9-s + 1.12·10-s − 1.03·11-s + 1.57·12-s + 3.78·13-s − 5.16·14-s − 2.15·15-s − 2.00·16-s + 17-s + 1.84·18-s + 4.34·19-s − 0.730·20-s + 9.87·21-s − 1.16·22-s + 1.72·23-s + 6.62·24-s + 25-s + 4.26·26-s + 2.93·27-s + 3.34·28-s + ⋯ |
| L(s) = 1 | + 0.796·2-s − 1.24·3-s − 0.365·4-s + 0.447·5-s − 0.990·6-s − 1.73·7-s − 1.08·8-s + 0.545·9-s + 0.356·10-s − 0.311·11-s + 0.453·12-s + 1.04·13-s − 1.38·14-s − 0.555·15-s − 0.501·16-s + 0.242·17-s + 0.434·18-s + 0.996·19-s − 0.163·20-s + 2.15·21-s − 0.248·22-s + 0.359·23-s + 1.35·24-s + 0.200·25-s + 0.835·26-s + 0.565·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 + 2.15T + 3T^{2} \) |
| 7 | \( 1 + 4.58T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 - 1.72T + 23T^{2} \) |
| 29 | \( 1 + 9.42T + 29T^{2} \) |
| 31 | \( 1 - 1.22T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 9.82T + 43T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 + 3.58T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 0.536T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 6.30T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.38729681378154426281569730312, −6.59073418909653824790159275213, −6.10726306415396563046376923787, −5.53459592492170411899300139576, −5.19088203457330436998435566650, −3.98867514019323622506797101078, −3.45857655574204095203105622469, −2.66070449451035434745969769613, −0.986175932531989886452512588242, 0,
0.986175932531989886452512588242, 2.66070449451035434745969769613, 3.45857655574204095203105622469, 3.98867514019323622506797101078, 5.19088203457330436998435566650, 5.53459592492170411899300139576, 6.10726306415396563046376923787, 6.59073418909653824790159275213, 7.38729681378154426281569730312
