| L(s) = 1 | − i·2-s − 2.30i·3-s − 4-s − 2.30·6-s + 0.697i·7-s + i·8-s − 2.30·9-s + 2.60·11-s + 2.30i·12-s − 6.30i·13-s + 0.697·14-s + 16-s − 3.90i·17-s + 2.30i·18-s + 0.605·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.32i·3-s − 0.5·4-s − 0.940·6-s + 0.263i·7-s + 0.353i·8-s − 0.767·9-s + 0.785·11-s + 0.664i·12-s − 1.74i·13-s + 0.186·14-s + 0.250·16-s − 0.947i·17-s + 0.542i·18-s + 0.138·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309832693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309832693\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2.30iT - 3T^{2} \) |
| 7 | \( 1 - 0.697iT - 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 6.30iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 0.605T + 19T^{2} \) |
| 23 | \( 1 + 1.69iT - 23T^{2} \) |
| 31 | \( 1 + 7.90T + 31T^{2} \) |
| 37 | \( 1 - 9.81iT - 37T^{2} \) |
| 41 | \( 1 + 8.60T + 41T^{2} \) |
| 43 | \( 1 + 3.30iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 3.90iT - 53T^{2} \) |
| 59 | \( 1 - 0.908T + 59T^{2} \) |
| 61 | \( 1 + 3.09T + 61T^{2} \) |
| 67 | \( 1 + 9.21iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2.51iT - 73T^{2} \) |
| 79 | \( 1 - 4.90T + 79T^{2} \) |
| 83 | \( 1 - 8.60iT - 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 1.09iT - 97T^{2} \) |
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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
−9.015686100543230877211949580449, −8.221963815101628701112138156002, −7.51067444877363075446262469372, −6.71475137899833044835584656319, −5.77008806569789362088313519445, −4.93636456837988367681870211964, −3.53108349246535564761866250355, −2.67922422659027618745887904629, −1.59044157146350117275570730785, −0.54303561979431377975275011123,
1.76028422211911830902300797787, 3.72912274742285476047244782898, 3.97573919137954430439863296118, 4.90232303695332888213484002994, 5.81860893541755335968935856479, 6.73498678801770435440978168179, 7.44303105679277718286835162693, 8.709761803594694154982180362000, 9.140567713647261771684257929340, 9.746205592861956623220914499623
