| L(s) = 1 | + 1.41i·5-s − i·7-s + 5.65·11-s + 4·13-s + 1.41i·17-s − 8i·19-s − 5.65·23-s + 2.99·25-s − 7.07i·29-s + 1.41·35-s − 2·37-s − 7.07i·41-s − 8i·43-s − 49-s + 9.89i·53-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s − 0.377i·7-s + 1.70·11-s + 1.10·13-s + 0.342i·17-s − 1.83i·19-s − 1.17·23-s + 0.599·25-s − 1.31i·29-s + 0.239·35-s − 0.328·37-s − 1.10i·41-s − 1.21i·43-s − 0.142·49-s + 1.35i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.168643032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.168643032\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 + 8iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−8.546357190019333267671361364644, −7.49025796833234854963440515714, −6.77934052458871416151837764807, −6.37610383663278315306194909753, −5.54824583372706696401391592506, −4.23567130977292447729165501816, −3.94176934696243463116602044745, −2.92382561087011068385504902401, −1.83700226315812534593332749843, −0.71085802053081255063795298137,
1.19071698470927665857351669318, 1.72930885926976960830708287200, 3.27829764032969069399782727756, 3.88321733571147502447004877214, 4.68210954926185838114855372776, 5.69350634602114769634069425739, 6.23264249004925874359560642903, 6.90943271888477644448187742274, 8.047308238963590468388660259562, 8.506343639494821649966739312178
