L(s) = 1 | + 3-s + (−1.67 − 1.67i)5-s + (1.16 − 1.16i)7-s + 9-s + (0.391 + 0.391i)11-s + (−3.59 − 0.311i)13-s + (−1.67 − 1.67i)15-s − 2.95i·17-s + (0.785 − 0.785i)19-s + (1.16 − 1.16i)21-s − 2.14·23-s + 0.591i·25-s + 27-s − 9.69i·29-s + (−1.16 − 1.16i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.747 − 0.747i)5-s + (0.438 − 0.438i)7-s + 0.333·9-s + (0.118 + 0.118i)11-s + (−0.996 − 0.0863i)13-s + (−0.431 − 0.431i)15-s − 0.717i·17-s + (0.180 − 0.180i)19-s + (0.253 − 0.253i)21-s − 0.447·23-s + 0.118i·25-s + 0.192·27-s − 1.80i·29-s + (−0.208 − 0.208i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.348435620\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348435620\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + (3.59 + 0.311i)T \) |
good | 5 | \( 1 + (1.67 + 1.67i)T + 5iT^{2} \) |
| 7 | \( 1 + (-1.16 + 1.16i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.391 - 0.391i)T + 11iT^{2} \) |
| 17 | \( 1 + 2.95iT - 17T^{2} \) |
| 19 | \( 1 + (-0.785 + 0.785i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 29 | \( 1 + 9.69iT - 29T^{2} \) |
| 31 | \( 1 + (1.16 + 1.16i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.87 + 1.87i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.21 - 1.21i)T - 41iT^{2} \) |
| 43 | \( 1 - 6.64iT - 43T^{2} \) |
| 47 | \( 1 + (5.19 - 5.19i)T - 47iT^{2} \) |
| 53 | \( 1 + 13.9iT - 53T^{2} \) |
| 59 | \( 1 + (8.86 + 8.86i)T + 59iT^{2} \) |
| 61 | \( 1 - 5.77iT - 61T^{2} \) |
| 67 | \( 1 + (-5.85 + 5.85i)T - 67iT^{2} \) |
| 71 | \( 1 + (5.33 + 5.33i)T + 71iT^{2} \) |
| 73 | \( 1 + (5.90 + 5.90i)T + 73iT^{2} \) |
| 79 | \( 1 - 1.82iT - 79T^{2} \) |
| 83 | \( 1 + (3.29 - 3.29i)T - 83iT^{2} \) |
| 89 | \( 1 + (7.48 + 7.48i)T + 89iT^{2} \) |
| 97 | \( 1 + (-4.94 + 4.94i)T - 97iT^{2} \) |
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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
−9.529931498247610370114433691492, −8.474943570699964814072480640933, −7.81124010040046773151725225079, −7.35589878149653385880662555651, −6.13377951676081460297548966599, −4.73644930098844318112921479683, −4.47146511904019415742072068350, −3.25787184551411615916917209015, −2.04279063928939828352881094759, −0.51893068629086243624367304446,
1.75081643614830688675496902219, 2.90521545773001216137055599457, 3.71050564515569268953201193121, 4.74157627950154060781260941302, 5.76856816598066565262535409445, 6.99500423409337363667171446314, 7.43751990169258419254294070116, 8.371170024727315267200024272734, 8.966620393133058413357163391056, 10.02824991539856300973978143558