| L(s) = 1 | − 2.23i·5-s + 4.50i·7-s − 16.7i·11-s + 3.28·13-s − 7.91i·17-s − 5.02i·19-s + (9.32 − 21.0i)23-s − 5.00·25-s − 46.6·29-s − 19.2·31-s + 10.0·35-s + 7.65i·37-s − 44.7·41-s + 33.2i·43-s − 34.0·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.643i·7-s − 1.52i·11-s + 0.252·13-s − 0.465i·17-s − 0.264i·19-s + (0.405 − 0.914i)23-s − 0.200·25-s − 1.60·29-s − 0.622·31-s + 0.287·35-s + 0.206i·37-s − 1.09·41-s + 0.774i·43-s − 0.724·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2410291317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2410291317\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-9.32 + 21.0i)T \) |
| good | 7 | \( 1 - 4.50iT - 49T^{2} \) |
| 11 | \( 1 + 16.7iT - 121T^{2} \) |
| 13 | \( 1 - 3.28T + 169T^{2} \) |
| 17 | \( 1 + 7.91iT - 289T^{2} \) |
| 19 | \( 1 + 5.02iT - 361T^{2} \) |
| 29 | \( 1 + 46.6T + 841T^{2} \) |
| 31 | \( 1 + 19.2T + 961T^{2} \) |
| 37 | \( 1 - 7.65iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 33.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 34.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 41.8T + 3.48e3T^{2} \) |
| 61 | \( 1 - 60.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 99.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 75.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 4.63T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 95.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 39.5iT - 9.40e3T^{2} \) |
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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
−8.025232947898504748191242066175, −7.07326777453125779625867800730, −6.26379627174880543460469499571, −5.55379358401939985814837430677, −5.02904804393956702070952137210, −3.91635822500360112234661857393, −3.16542430634963832250296185803, −2.24909754768722089282106883867, −1.07063671649107605732449044810, −0.05143136385854800840368624402,
1.48929084647172952547463761041, 2.16679730816757670684576849314, 3.48508907308804473135970288733, 3.95556481574615986478018005209, 4.94030024382813178347955585805, 5.67788402089187261092716899493, 6.65668611084594382825894988920, 7.26357566068923210670157389533, 7.68023140809332179863374669712, 8.641027213255242190293329533976
