| L(s) = 1 | + i·2-s − 4-s + i·5-s + 1.48i·7-s − i·8-s − 10-s − 4.89i·11-s + (1.44 − 3.30i)13-s − 1.48·14-s + 16-s + 8.08·17-s + 5.11i·19-s − i·20-s + 4.89·22-s + 1.48·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.447i·5-s + 0.560i·7-s − 0.353i·8-s − 0.316·10-s − 1.47i·11-s + (0.402 − 0.915i)13-s − 0.396·14-s + 0.250·16-s + 1.96·17-s + 1.17i·19-s − 0.223i·20-s + 1.04·22-s + 0.309·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.638458059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638458059\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-1.44 + 3.30i)T \) |
| good | 7 | \( 1 - 1.48iT - 7T^{2} \) |
| 11 | \( 1 + 4.89iT - 11T^{2} \) |
| 17 | \( 1 - 8.08T + 17T^{2} \) |
| 19 | \( 1 - 5.11iT - 19T^{2} \) |
| 23 | \( 1 - 1.48T + 23T^{2} \) |
| 29 | \( 1 - 5.11T + 29T^{2} \) |
| 31 | \( 1 - 6.60iT - 31T^{2} \) |
| 37 | \( 1 - 3.63iT - 37T^{2} \) |
| 41 | \( 1 - 1.10iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 9.79iT - 47T^{2} \) |
| 53 | \( 1 + 6.60T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 9.57iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4.45iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 9.79iT - 83T^{2} \) |
| 89 | \( 1 + 10.8iT - 89T^{2} \) |
| 97 | \( 1 + 17.6iT - 97T^{2} \) |
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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
−10.10242597887074153121363699504, −8.775720442513972075126723835908, −8.275503228832058249329107885537, −7.58806060696614144363163530815, −6.42495109713175980124191742670, −5.75503367352555655545856588784, −5.20748363446171632546744038248, −3.54918781415330901874275284304, −3.07718227381509566250502320187, −1.08236274561296325687845051201,
0.972462739556301515260437863268, 2.10533924539346724359435253838, 3.41112504413909191662494746535, 4.43089550595069608157075082962, 4.99010638410824823585093267322, 6.26877889794645567525149814801, 7.32369122012630293252627182625, 7.959482318573225319103001744767, 9.178071108150290148390293930135, 9.598695088855448828973988670640
