| L(s) = 1 | + (0.207 − 0.978i)2-s + (0.809 − 0.587i)3-s + (−0.913 − 0.406i)4-s + (−0.406 + 0.913i)5-s + (−0.406 − 0.913i)6-s + (−0.866 − 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (0.604 + 0.128i)11-s + (−0.978 + 0.207i)12-s + (−0.336 − 1.58i)13-s + (−0.669 + 0.743i)14-s + (0.207 + 0.978i)15-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
| L(s) = 1 | + (0.207 − 0.978i)2-s + (0.809 − 0.587i)3-s + (−0.913 − 0.406i)4-s + (−0.406 + 0.913i)5-s + (−0.406 − 0.913i)6-s + (−0.866 − 0.5i)7-s + (−0.587 + 0.809i)8-s + (0.309 − 0.951i)9-s + (0.809 + 0.587i)10-s + (0.604 + 0.128i)11-s + (−0.978 + 0.207i)12-s + (−0.336 − 1.58i)13-s + (−0.669 + 0.743i)14-s + (0.207 + 0.978i)15-s + (0.669 + 0.743i)16-s + (−0.809 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.047661309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047661309\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (0.406 - 0.913i)T \) |
| good | 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.604 - 0.128i)T + (0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.336 + 1.58i)T + (-0.913 + 0.406i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.743 + 0.669i)T + (0.104 + 0.994i)T^{2} \) |
| 29 | \( 1 + (-0.614 + 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-1.58 + 0.336i)T + (0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 61 | \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \) |
| 67 | \( 1 + (0.169 - 1.60i)T + (-0.978 - 0.207i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.614 - 0.0646i)T + (0.978 - 0.207i)T^{2} \) |
| 83 | \( 1 + (-0.564 + 0.251i)T + (0.669 - 0.743i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
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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
−9.213045177873701217670138299007, −8.377618318535007875934857708536, −7.62497057433924395507930582390, −6.67200840912887389296683816181, −6.13871269550395831840598914747, −4.53954607183980206653741083883, −3.78709737255910691061524252919, −2.86963483958751385997363460974, −2.45751851769454879973491021913, −0.65787479445534311836857715219,
2.02193224799647274025422948101, 3.51048013425426363447693155273, 4.20578577615222657163917852115, 4.70301509100674632861968531469, 5.93843669275691616154422410598, 6.59389886971813200037069872030, 7.57081672821671703803960974335, 8.524023452907433907597619345052, 8.810009667984859065319476163759, 9.503170520082510548853970251582
