| L(s) = 1 | − 4i·2-s − 3i·3-s − 8·4-s − 12·6-s + 7i·7-s − 9·9-s + 62·11-s + 24i·12-s − 62i·13-s + 28·14-s − 64·16-s − 84i·17-s + 36i·18-s − 100·19-s + 21·21-s − 248i·22-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s − 0.577i·3-s − 4-s − 0.816·6-s + 0.377i·7-s − 0.333·9-s + 1.69·11-s + 0.577i·12-s − 1.32i·13-s + 0.534·14-s − 16-s − 1.19i·17-s + 0.471i·18-s − 1.20·19-s + 0.218·21-s − 2.40i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.412618549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412618549\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
| good | 2 | \( 1 + 4iT - 8T^{2} \) |
| 11 | \( 1 - 62T + 1.33e3T^{2} \) |
| 13 | \( 1 + 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 84iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 100T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 10T + 2.43e4T^{2} \) |
| 31 | \( 1 + 48T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 248T + 6.89e4T^{2} \) |
| 43 | \( 1 - 68iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 324iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 258iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 120T + 2.05e5T^{2} \) |
| 61 | \( 1 - 622T + 2.26e5T^{2} \) |
| 67 | \( 1 + 904iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 678T + 3.57e5T^{2} \) |
| 73 | \( 1 + 642iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 740T + 4.93e5T^{2} \) |
| 83 | \( 1 - 468iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 200T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3iT - 9.12e5T^{2} \) |
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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
−10.01636620284780058560548823370, −9.136627175657946579941832332024, −8.380191019279002068946181413832, −7.02725612334736470774967569769, −6.21884620100085615494131017259, −4.82227763711005961380477982938, −3.60999989998045946874408501070, −2.66305426886680490382078971294, −1.53388401259014487712243343486, −0.43175895752652985751343205627,
1.78704493933456862957921465581, 3.92720484187089185459946482951, 4.38873709311673431788596034395, 5.78781640174166535767489222834, 6.53676720341667882133822843637, 7.16649272725830390843696050666, 8.508379259442428141194372983117, 8.916941971575279439224875577701, 9.891277397216770320676781197489, 11.06599068557064614644669511188
