L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.951 + 0.309i)3-s + (0.809 − 0.587i)4-s + (2.21 − 0.332i)5-s − 0.999·6-s + (−0.907 − 1.24i)7-s + (−0.587 + 0.809i)8-s + (0.809 + 0.587i)9-s + (−2.00 + 0.999i)10-s + (1.74 − 1.27i)11-s + (0.951 − 0.309i)12-s + (0.363 + 0.118i)13-s + (1.24 + 0.907i)14-s + (2.20 + 0.367i)15-s + (0.309 − 0.951i)16-s + (−0.638 + 0.879i)17-s + ⋯ |
L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.549 + 0.178i)3-s + (0.404 − 0.293i)4-s + (0.988 − 0.148i)5-s − 0.408·6-s + (−0.342 − 0.471i)7-s + (−0.207 + 0.286i)8-s + (0.269 + 0.195i)9-s + (−0.632 + 0.315i)10-s + (0.527 − 0.383i)11-s + (0.274 − 0.0892i)12-s + (0.100 + 0.0327i)13-s + (0.333 + 0.242i)14-s + (0.569 + 0.0948i)15-s + (0.0772 − 0.237i)16-s + (−0.154 + 0.213i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69927 - 0.0162958i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69927 - 0.0162958i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.21 + 0.332i)T \) |
| 31 | \( 1 + (-5.47 + 1.01i)T \) |
good | 7 | \( 1 + (0.907 + 1.24i)T + (-2.16 + 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.74 + 1.27i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.363 - 0.118i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.638 - 0.879i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.32 - 4.07i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.62 + 3.61i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (1.31 + 4.05i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 - 3.70iT - 37T^{2} \) |
| 41 | \( 1 + (-0.587 - 1.80i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-7.37 + 2.39i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (9.01 + 2.92i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.47 - 4.78i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.08 + 3.33i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + 4.97T + 61T^{2} \) |
| 67 | \( 1 + 2.20iT - 67T^{2} \) |
| 71 | \( 1 + (4.31 + 3.13i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.86 - 5.32i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.01 - 2.91i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.57 + 0.512i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.341 - 0.248i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.18 - 11.2i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−9.968696241467009386206276869321, −9.246634211328836763550963578743, −8.538328017705094465383154903452, −7.70765429066027540574833054265, −6.58608800024629245374314041128, −6.06065416903890276257594000583, −4.81435420728876564977723591619, −3.57670165840534032856495567183, −2.38463666876167301779126593242, −1.13846250617316543489850696159,
1.33160153826247365721837310942, 2.45492731578053504680816572044, 3.27000575376514930390129295669, 4.79387960245374624862144133114, 5.98193237504908822618249028073, 6.80314647584084296106933038687, 7.53829993050422790328049880246, 8.759376609800249569739815348067, 9.260454904204058646978585308247, 9.761442010979827369227799451311