| L(s) = 1 | + 10.1i·5-s − 33.8·7-s − 13.5i·11-s − 45.9i·13-s − 131.·17-s + 7.89i·19-s − 28.2·23-s + 21.9·25-s + 238. i·29-s − 67.2·31-s − 343. i·35-s + 80.2i·37-s − 299.·41-s − 378. i·43-s − 367.·47-s + ⋯ |
| L(s) = 1 | + 0.907i·5-s − 1.82·7-s − 0.372i·11-s − 0.979i·13-s − 1.87·17-s + 0.0953i·19-s − 0.256·23-s + 0.175·25-s + 1.52i·29-s − 0.389·31-s − 1.65i·35-s + 0.356i·37-s − 1.14·41-s − 1.34i·43-s − 1.14·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7662584013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7662584013\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 10.1iT - 125T^{2} \) |
| 7 | \( 1 + 33.8T + 343T^{2} \) |
| 11 | \( 1 + 13.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 45.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 131.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.89iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 28.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 238. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 67.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 80.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 299.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 378. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 367.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 28.5iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 597. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 142. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 656. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 982.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 900.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 650. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.23e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 25.6T + 9.12e5T^{2} \) |
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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
−9.058642396435785526632816547355, −8.301501382241380416304733314234, −7.01201330383553190453207270118, −6.73053174334540721048916300278, −6.00941678309464603019056703011, −4.95820088517076007465369595901, −3.52012246453326810248108713782, −3.22218900412448985735541774730, −2.18542918755418707013748107829, −0.38378468946026464039195634919,
0.40001451728242619346470265640, 1.87419151297015742475889378184, 2.88521017633911011432028788009, 4.08803785460288741866889833743, 4.59138442836635133485350334823, 5.84385701346906724329337882708, 6.58052890776412828808885272588, 7.08696331643813927029178325295, 8.368558862091263060362283001919, 9.077012054752415411542810323989
