| L(s) = 1 | − 5.19i·3-s − 19.7·5-s − 14.6i·7-s − 27·9-s − 72.7i·11-s + 102. i·15-s − 76.3·21-s + 267·25-s + 140. i·27-s − 223.·29-s + 338. i·31-s − 378·33-s + 290. i·35-s + 534.·45-s + 127·49-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s − 1.77·5-s − 0.793i·7-s − 9-s − 1.99i·11-s + 1.77i·15-s − 0.793·21-s + 2.13·25-s + 1.00i·27-s − 1.43·29-s + 1.95i·31-s − 1.99·33-s + 1.40i·35-s + 1.77·45-s + 0.370·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1743976282\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1743976282\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 + 14.6iT - 343T^{2} \) |
| 11 | \( 1 + 72.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 338. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 579.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 717. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 322T + 3.89e5T^{2} \) |
| 79 | \( 1 + 308. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 883. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 + 574T + 9.12e5T^{2} \) |
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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
−10.74445904676533284998543789778, −8.817362508683470407579795774750, −8.241118366546835072501454063215, −7.45197788894682430086376673578, −6.69330962535993420628985677129, −5.40717127083419086611169310767, −3.84726455468621692537444513998, −3.14810597289507400135949194956, −0.992316993639454409652799630127, −0.07475387363303661820435515626,
2.46469733520374355879346299386, 3.90623325847472728747429711299, 4.44256206640290194843437827735, 5.56883377452227372291410318242, 7.16315580762491008990511177209, 7.894089727863547590176517569708, 8.938947251886610586997740707914, 9.710435419844413537705458091655, 10.75273315544768636971862436215, 11.74006791432998509351435999800
