| L(s) = 1 | + 7i·3-s + (10 − 5i)5-s + 7i·7-s − 22·9-s + 37·11-s + 51i·13-s + (35 + 70i)15-s − 41i·17-s − 108·19-s − 49·21-s + 70i·23-s + (75 − 100i)25-s + 35i·27-s + 249·29-s + 134·31-s + ⋯ |
| L(s) = 1 | + 1.34i·3-s + (0.894 − 0.447i)5-s + 0.377i·7-s − 0.814·9-s + 1.01·11-s + 1.08i·13-s + (0.602 + 1.20i)15-s − 0.584i·17-s − 1.30·19-s − 0.509·21-s + 0.634i·23-s + (0.599 − 0.800i)25-s + 0.249i·27-s + 1.59·29-s + 0.776·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.341488155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.341488155\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-10 + 5i)T \) |
| 7 | \( 1 - 7iT \) |
| good | 3 | \( 1 - 7iT - 27T^{2} \) |
| 11 | \( 1 - 37T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 41iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 108T + 6.85e3T^{2} \) |
| 23 | \( 1 - 70iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 249T + 2.43e4T^{2} \) |
| 31 | \( 1 - 134T + 2.97e4T^{2} \) |
| 37 | \( 1 - 334iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 206T + 6.89e4T^{2} \) |
| 43 | \( 1 - 376iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 287iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 6iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 940T + 2.26e5T^{2} \) |
| 67 | \( 1 - 106iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 456T + 3.57e5T^{2} \) |
| 73 | \( 1 - 650iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.23e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 428iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 220T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3iT - 9.12e5T^{2} \) |
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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
−10.36926524053642994347746766046, −9.711539885922383199471364738682, −9.090418139899520724799932438143, −8.473580210228350615918452670062, −6.72509932220395528765474557824, −6.01127898420768688588150096950, −4.72170020670920079599701554482, −4.34504249616543032403505156155, −2.87648906640323879830156879710, −1.46430282452625103504691213810,
0.73344971104536451736237223283, 1.80617023876553434604988669613, 2.84258557771762936701777648102, 4.34761980199088563828425839718, 5.95542838875982992346702501999, 6.39729741847249942655004039436, 7.22032090889401247138986442513, 8.182319013789018561247918311934, 9.050852936863195861484205665299, 10.32136851244820850209315865250
