L(s) = 1 | − 2-s + (1.29 + 1.29i)3-s + 4-s + (−2.20 − 0.390i)5-s + (−1.29 − 1.29i)6-s + (−2.67 − 2.67i)7-s − 8-s + 0.348i·9-s + (2.20 + 0.390i)10-s − 1.29i·11-s + (1.29 + 1.29i)12-s − 6.92·13-s + (2.67 + 2.67i)14-s + (−2.34 − 3.35i)15-s + 16-s − 4.85i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.747 + 0.747i)3-s + 0.5·4-s + (−0.984 − 0.174i)5-s + (−0.528 − 0.528i)6-s + (−1.00 − 1.00i)7-s − 0.353·8-s + 0.116i·9-s + (0.696 + 0.123i)10-s − 0.390i·11-s + (0.373 + 0.373i)12-s − 1.92·13-s + (0.713 + 0.713i)14-s + (−0.604 − 0.866i)15-s + 0.250·16-s − 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.158682 - 0.307190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.158682 - 0.307190i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 + (2.20 + 0.390i)T \) |
| 37 | \( 1 + (6.03 + 0.770i)T \) |
good | 3 | \( 1 + (-1.29 - 1.29i)T + 3iT^{2} \) |
| 7 | \( 1 + (2.67 + 2.67i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.29iT - 11T^{2} \) |
| 13 | \( 1 + 6.92T + 13T^{2} \) |
| 17 | \( 1 + 4.85iT - 17T^{2} \) |
| 19 | \( 1 + (4.69 - 4.69i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 + (-3.73 - 3.73i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.146 - 0.146i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.38iT - 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 + (2.93 + 2.93i)T + 47iT^{2} \) |
| 53 | \( 1 + (1.30 - 1.30i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.70 + 4.70i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.81 - 7.81i)T - 61iT^{2} \) |
| 67 | \( 1 + (-3.10 + 3.10i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + (-0.134 - 0.134i)T + 73iT^{2} \) |
| 79 | \( 1 + (-10.8 + 10.8i)T - 79iT^{2} \) |
| 83 | \( 1 + (8.74 - 8.74i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.83 - 3.83i)T + 89iT^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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
−10.67093791088449964915085463143, −10.05881508277880576063205628789, −9.293373774187147197675687474942, −8.451859588618355487009110115522, −7.38181953293403024196672461121, −6.78286425022798306615591831970, −4.85508606736216611756263390128, −3.73906245586255080400870195419, −2.88498273281120103866781399386, −0.24753968932245616152820951048,
2.26712868570617692916936398062, 3.01164337525710532415074659915, 4.76931144375445609442218723334, 6.50269040149944890498138471144, 7.15166049354468097997930604618, 8.081251813532202067469901951649, 8.769959370569853141348326092358, 9.690985747006468718291776179314, 10.68833981198372664775142537212, 11.89932071701319869342548865760