L(s) = 1 | + (1.87 − 1.87i)3-s + (−1.41 + 1.41i)7-s − 4i·9-s + 5.29i·11-s + (1.87 − 1.87i)13-s + 5.29·19-s + 5.29i·21-s + (0.957 + 4.69i)23-s + (−1.87 − 1.87i)27-s + i·29-s + 3·31-s + (9.89 + 9.89i)33-s + (7.07 − 7.07i)37-s − 7i·39-s + 9·41-s + ⋯ |
L(s) = 1 | + (1.08 − 1.08i)3-s + (−0.534 + 0.534i)7-s − 1.33i·9-s + 1.59i·11-s + (0.518 − 0.518i)13-s + 1.21·19-s + 1.15i·21-s + (0.199 + 0.979i)23-s + (−0.360 − 0.360i)27-s + 0.185i·29-s + 0.538·31-s + (1.72 + 1.72i)33-s + (1.16 − 1.16i)37-s − 1.12i·39-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.637068906\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.637068906\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-0.957 - 4.69i)T \) |
good | 3 | \( 1 + (-1.87 + 1.87i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.41 - 1.41i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.29iT - 11T^{2} \) |
| 13 | \( 1 + (-1.87 + 1.87i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 29 | \( 1 - iT - 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + (-7.07 + 7.07i)T - 37iT^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + (4.24 + 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.87 + 1.87i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 5.29iT - 61T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (-9.35 + 9.35i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + (-1.41 + 1.41i)T - 97iT^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.087658124755222198052899740507, −7.935040906005482710636703044245, −7.57652395271108286913932840597, −6.87967359962724506365155514943, −6.00338794199987178252513132316, −5.06254965884393502083720484057, −3.81435575515360248779456649049, −2.92779798875699446705692387366, −2.18449062216632006279645032143, −1.16895900925532800959835261838,
0.951093365315849331947592801084, 2.74236224879773729351677789384, 3.26843363156486795727331908212, 4.02115758975093772207136491423, 4.80590231714070933935978262587, 5.95973792544123700893523592153, 6.65687055575892683207122646676, 7.87015153378100251521540490538, 8.369001847408232100997695930410, 9.125297386306260065533407026241