| L(s) = 1 | − i·3-s − 3.69i·5-s − 7-s − 9-s + 3.21i·11-s − 5.08i·13-s − 3.69·15-s + 0.616·17-s − 4.48i·19-s + i·21-s − 1.38·23-s − 8.67·25-s + i·27-s + 5.67i·29-s − 6.91·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.65i·5-s − 0.377·7-s − 0.333·9-s + 0.968i·11-s − 1.40i·13-s − 0.954·15-s + 0.149·17-s − 1.02i·19-s + 0.218i·21-s − 0.288·23-s − 1.73·25-s + 0.192i·27-s + 1.05i·29-s − 1.24·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.241884 - 1.04504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241884 - 1.04504i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3.69iT - 5T^{2} \) |
| 11 | \( 1 - 3.21iT - 11T^{2} \) |
| 13 | \( 1 + 5.08iT - 13T^{2} \) |
| 17 | \( 1 - 0.616T + 17T^{2} \) |
| 19 | \( 1 + 4.48iT - 19T^{2} \) |
| 23 | \( 1 + 1.38T + 23T^{2} \) |
| 29 | \( 1 - 5.67iT - 29T^{2} \) |
| 31 | \( 1 + 6.91T + 31T^{2} \) |
| 37 | \( 1 - 6.91iT - 37T^{2} \) |
| 41 | \( 1 + 0.616T + 41T^{2} \) |
| 43 | \( 1 + 7.99iT - 43T^{2} \) |
| 47 | \( 1 + 4.97T + 47T^{2} \) |
| 53 | \( 1 + 4.48iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 12.4iT - 61T^{2} \) |
| 67 | \( 1 + 9.56iT - 67T^{2} \) |
| 71 | \( 1 - 15.2T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 10.4iT - 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 9.73T + 97T^{2} \) |
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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
−9.951487539775736761832626115598, −9.194661143377129458933079378103, −8.391107508057900772343522295101, −7.63512883470481700051397519940, −6.63176531719125770704253100855, −5.35240212362635618058292256400, −4.89609927693205567021358196923, −3.45050967869519113768929490690, −1.90183814289511450978458086845, −0.55789665340972688815260148631,
2.25629264749389193187153452818, 3.40989650807353022230008631612, 4.06105053458875008439726220617, 5.72517025042262263691196890061, 6.37375022321200336871555247898, 7.24786857348956595256755548400, 8.267665302142243675138732371213, 9.432895230794566173279407221565, 9.989540036357307050124259422135, 11.03694769819307509823106948425
