| L(s) = 1 | + (−2.08 − 0.806i)5-s + 2i·7-s − 5.17·11-s − 1.61i·13-s − 3.61i·17-s + 1.17·19-s − 8.34i·23-s + (3.69 + 3.36i)25-s + 6.22·29-s − 1.94·31-s + (1.61 − 4.17i)35-s + 7.95i·37-s − 0.0554·41-s − 6i·43-s + 6.34i·47-s + ⋯ |
| L(s) = 1 | + (−0.932 − 0.360i)5-s + 0.755i·7-s − 1.55·11-s − 0.447i·13-s − 0.876i·17-s + 0.268·19-s − 1.73i·23-s + (0.739 + 0.672i)25-s + 1.15·29-s − 0.349·31-s + (0.272 − 0.705i)35-s + 1.30i·37-s − 0.00865·41-s − 0.914i·43-s + 0.925i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8453205449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8453205449\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.08 + 0.806i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + 1.61iT - 13T^{2} \) |
| 17 | \( 1 + 3.61iT - 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 8.34iT - 23T^{2} \) |
| 29 | \( 1 - 6.22T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 - 7.95iT - 37T^{2} \) |
| 41 | \( 1 + 0.0554T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 6.34iT - 47T^{2} \) |
| 53 | \( 1 - 14.3iT - 53T^{2} \) |
| 59 | \( 1 + 4.39T + 59T^{2} \) |
| 61 | \( 1 + 4.17T + 61T^{2} \) |
| 67 | \( 1 - 8.34iT - 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 - 7.61iT - 73T^{2} \) |
| 79 | \( 1 - 3.22T + 79T^{2} \) |
| 83 | \( 1 - 14.3iT - 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 10.3iT - 97T^{2} \) |
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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
−8.627725484307270155044682302176, −8.098864906687738779304569773594, −7.49405478470370165981623114189, −6.61299554204423353800695707466, −5.58318484773677475857881350882, −4.96172784503671305032362817071, −4.30958606218072126837763631870, −2.92717876749038796235720636502, −2.62037981694272815832366673407, −0.848326457770095931771327027690,
0.34474713180030522260592163009, 1.84027433793658796684028887767, 3.07649059243477572696248733356, 3.71746817582364378376621287051, 4.58461818112971763108027455365, 5.38408219113148757642599700378, 6.35007462364688103219507571711, 7.28542116530408944044608470959, 7.67128609063741966462830528039, 8.282090467960650258557949941903
