| L(s) = 1 | + 2.56·3-s + (2 + i)5-s + (2 + 1.73i)7-s + 3.56·9-s − 0.972·11-s − 0.561i·13-s + (5.12 + 2.56i)15-s − 4.43·17-s + 1.12i·19-s + (5.12 + 4.43i)21-s − 1.12·23-s + (3 + 4i)25-s + 1.43·27-s + 4.43i·29-s + 8.87·31-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + (0.894 + 0.447i)5-s + (0.755 + 0.654i)7-s + 1.18·9-s − 0.293·11-s − 0.155i·13-s + (1.32 + 0.661i)15-s − 1.07·17-s + 0.257i·19-s + (1.11 + 0.968i)21-s − 0.234·23-s + (0.600 + 0.800i)25-s + 0.276·27-s + 0.823i·29-s + 1.59·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.776085897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.776085897\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 11 | \( 1 + 0.972T + 11T^{2} \) |
| 13 | \( 1 + 0.561iT - 13T^{2} \) |
| 17 | \( 1 + 4.43T + 17T^{2} \) |
| 19 | \( 1 - 1.12iT - 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43iT - 29T^{2} \) |
| 31 | \( 1 - 8.87T + 31T^{2} \) |
| 37 | \( 1 - 8.87T + 37T^{2} \) |
| 41 | \( 1 + 8.87iT - 41T^{2} \) |
| 43 | \( 1 - 1.94iT - 43T^{2} \) |
| 47 | \( 1 + 0.972iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 + 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 14.2iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 7.68iT - 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 - 8.87iT - 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−9.108406414749319868285225416011, −8.364436207224638629974934039338, −7.85017685029248234696302076532, −6.88173667304686443531578346824, −6.03082669566661275193352577306, −5.09964505673034529432087093330, −4.15903480933583414242058057113, −2.95523824538860828368460475055, −2.42604407419008924533586622194, −1.62848863576688614597279371286,
1.17129284194954539009035775710, 2.23106394785741086964227037376, 2.82876199701959423177932092728, 4.25822748984706555415454154919, 4.58395942987409979218539456945, 5.84285301127154471372125717639, 6.75660410287918487855671806044, 7.67360337397996141026060073651, 8.331896560647272841886293883228, 8.785117452009476897766321741044
