L(s) = 1 | + (1.58 + 1.58i)5-s + (1.23 − 1.23i)7-s + 1.74i·11-s + (−0.236 − 0.236i)13-s + (−4.57 − 4.57i)17-s + 6.47i·19-s + (2.82 − 2.82i)23-s + 5.00i·25-s + 0.333·29-s + 10.4·31-s + 3.90·35-s + (2.23 − 2.23i)37-s + 7.07i·41-s + (6.47 + 6.47i)43-s + (4.57 + 4.57i)47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s + (0.467 − 0.467i)7-s + 0.527i·11-s + (−0.0654 − 0.0654i)13-s + (−1.10 − 1.10i)17-s + 1.48i·19-s + (0.589 − 0.589i)23-s + 1.00i·25-s + 0.0619·29-s + 1.88·31-s + 0.660·35-s + (0.367 − 0.367i)37-s + 1.10i·41-s + (0.986 + 0.986i)43-s + (0.667 + 0.667i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.167075033\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167075033\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 - 1.58i)T \) |
good | 7 | \( 1 + (-1.23 + 1.23i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.74iT - 11T^{2} \) |
| 13 | \( 1 + (0.236 + 0.236i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.57 + 4.57i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.47iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.333T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + (-2.23 + 2.23i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-6.47 - 6.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.57 - 4.57i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 + 7.40T + 59T^{2} \) |
| 61 | \( 1 + 1.52T + 61T^{2} \) |
| 67 | \( 1 + (-10.4 + 10.4i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (9.47 + 9.47i)T + 73iT^{2} \) |
| 79 | \( 1 + 5.52iT - 79T^{2} \) |
| 83 | \( 1 + (7.40 - 7.40i)T - 83iT^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−8.996479212475293613823690119899, −7.942705218767524047789745482449, −7.40748277366802892246220002681, −6.51724674491083937816704773343, −6.02826228698678785148384386983, −4.84043997338236697847418111440, −4.33767538674694827713872044391, −3.02967353794630382638203078242, −2.31352081332374865945550401254, −1.16413781837659209845111547137,
0.77273131687328518252220010391, 1.97569514414404362287309577312, 2.74193491351231495445569640391, 4.10664538304590534779273295065, 4.83954408235061579114014380254, 5.56100433092002381619494758978, 6.30457230718299759952955783649, 7.06342402643362123586158849974, 8.190174257075107087555345628830, 8.798250199249701883075576243370