| L(s) = 1 | + (−0.415 + 0.909i)3-s + (2.10 + 0.618i)5-s + (0.474 − 0.0914i)7-s + (−0.654 − 0.755i)9-s + (3.02 − 2.88i)11-s + (0.287 − 6.04i)13-s + (−1.43 + 1.65i)15-s + (0.106 − 0.0833i)17-s + (4.57 + 0.882i)19-s + (−0.113 + 0.469i)21-s + (0.0590 − 0.00564i)23-s + (−0.148 − 0.0952i)25-s + (0.959 − 0.281i)27-s + (−0.202 + 0.350i)29-s + (0.214 + 4.49i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (0.942 + 0.276i)5-s + (0.179 − 0.0345i)7-s + (−0.218 − 0.251i)9-s + (0.911 − 0.868i)11-s + (0.0798 − 1.67i)13-s + (−0.371 + 0.428i)15-s + (0.0257 − 0.0202i)17-s + (1.04 + 0.202i)19-s + (−0.0248 + 0.102i)21-s + (0.0123 − 0.00117i)23-s + (−0.0296 − 0.0190i)25-s + (0.184 − 0.0542i)27-s + (−0.0376 + 0.0651i)29-s + (0.0384 + 0.807i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80459 + 0.0476164i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80459 + 0.0476164i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (-1.95 - 7.94i)T \) |
| good | 5 | \( 1 + (-2.10 - 0.618i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.474 + 0.0914i)T + (6.49 - 2.60i)T^{2} \) |
| 11 | \( 1 + (-3.02 + 2.88i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.287 + 6.04i)T + (-12.9 - 1.23i)T^{2} \) |
| 17 | \( 1 + (-0.106 + 0.0833i)T + (4.00 - 16.5i)T^{2} \) |
| 19 | \( 1 + (-4.57 - 0.882i)T + (17.6 + 7.06i)T^{2} \) |
| 23 | \( 1 + (-0.0590 + 0.00564i)T + (22.5 - 4.35i)T^{2} \) |
| 29 | \( 1 + (0.202 - 0.350i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.214 - 4.49i)T + (-30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (1.16 + 2.01i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.08 - 0.833i)T + (29.6 + 28.2i)T^{2} \) |
| 43 | \( 1 + (-1.26 - 8.79i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (-0.663 - 0.931i)T + (-15.3 + 44.4i)T^{2} \) |
| 53 | \( 1 + (-0.919 + 6.39i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-8.18 + 5.25i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.09 - 2.94i)T + (2.90 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-4.67 - 3.67i)T + (16.7 + 68.9i)T^{2} \) |
| 73 | \( 1 + (5.09 + 4.85i)T + (3.47 + 72.9i)T^{2} \) |
| 79 | \( 1 + (6.81 - 3.51i)T + (45.8 - 64.3i)T^{2} \) |
| 83 | \( 1 + (1.19 + 4.92i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (-1.25 - 2.75i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.47 - 9.49i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.17260496903007729587046738406, −9.634246623367230963874976544352, −8.677438449547244102962457140379, −7.79504180613073189642026054301, −6.54827084618623853790496989816, −5.76997461221368886717215648801, −5.14850926870770491259383660167, −3.70792679279486642949165986221, −2.83432590081257829733787387062, −1.12069689842731627297871883658,
1.42901684032352195363066762068, 2.20583821819557632024572654680, 3.95637726682608418503518271083, 4.97133647425016578121787103267, 5.95122685913259573979870728977, 6.78317216312828005192678575499, 7.44479222557670806450859560637, 8.773764152498571613860476294445, 9.409164523646957050662891541541, 10.02610016109524188286668128495
