L(s) = 1 | + (2.03 − 0.929i)5-s + (2.49 + 2.49i)7-s + 3.92·11-s + (4.55 + 4.55i)13-s + (−1.88 + 1.88i)17-s − 4.61·19-s + (−0.741 − 0.741i)23-s + (3.27 − 3.78i)25-s + 4.35i·29-s − 9.67·31-s + (7.39 + 2.75i)35-s + (−5.39 + 5.39i)37-s − 6.33i·41-s + (−0.206 − 0.206i)43-s + (3.48 − 3.48i)47-s + ⋯ |
L(s) = 1 | + (0.909 − 0.415i)5-s + (0.942 + 0.942i)7-s + 1.18·11-s + (1.26 + 1.26i)13-s + (−0.457 + 0.457i)17-s − 1.05·19-s + (−0.154 − 0.154i)23-s + (0.654 − 0.756i)25-s + 0.809i·29-s − 1.73·31-s + (1.24 + 0.465i)35-s + (−0.887 + 0.887i)37-s − 0.989i·41-s + (−0.0314 − 0.0314i)43-s + (0.507 − 0.507i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.380525233\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.380525233\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.03 + 0.929i)T \) |
good | 7 | \( 1 + (-2.49 - 2.49i)T + 7iT^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + (-4.55 - 4.55i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.88 - 1.88i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.61T + 19T^{2} \) |
| 23 | \( 1 + (0.741 + 0.741i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.35iT - 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 + (5.39 - 5.39i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.33iT - 41T^{2} \) |
| 43 | \( 1 + (0.206 + 0.206i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.48 + 3.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.01 + 1.01i)T - 53iT^{2} \) |
| 59 | \( 1 + 0.531iT - 59T^{2} \) |
| 61 | \( 1 - 3.00iT - 61T^{2} \) |
| 67 | \( 1 + (-1.28 + 1.28i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.61iT - 71T^{2} \) |
| 73 | \( 1 + (0.509 - 0.509i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.31iT - 79T^{2} \) |
| 83 | \( 1 + (-9.85 + 9.85i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.91T + 89T^{2} \) |
| 97 | \( 1 + (8.11 + 8.11i)T + 97iT^{2} \) |
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
−9.197741217882916364267328974705, −8.857264602950313343570380374981, −8.458702677789087914461056945250, −6.93072175147327049694359664001, −6.27844676736525990716911856817, −5.54594170794185043884842809043, −4.57597068409972231695687081209, −3.71904472586850539334528436245, −1.94583959274105495149812734217, −1.65478176293444197562739949179,
1.07617566062421137725907104084, 2.06861080810491473634640150777, 3.49580922608865547669316403280, 4.26206550818307923561237616528, 5.41983181260277321360836825906, 6.18971232905169329465050085270, 6.96425586616244759815069231415, 7.83999040154943044802804044342, 8.709570662251242778845186138956, 9.425192552862526948776531239605