| L(s) = 1 | + 1.24i·3-s + 3.14·5-s + i·7-s + 1.45·9-s − 0.520i·11-s + 1.99i·13-s + 3.91i·15-s − 6.73i·17-s + 5.18i·19-s − 1.24·21-s + 8.85·23-s + 4.89·25-s + 5.54i·27-s − 1.05i·29-s + 0.606·31-s + ⋯ |
| L(s) = 1 | + 0.718i·3-s + 1.40·5-s + 0.377i·7-s + 0.483·9-s − 0.156i·11-s + 0.554i·13-s + 1.01i·15-s − 1.63i·17-s + 1.19i·19-s − 0.271·21-s + 1.84·23-s + 0.978·25-s + 1.06i·27-s − 0.195i·29-s + 0.108·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258559302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258559302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| 41 | \( 1 + (5.11 + 3.84i)T \) |
| good | 3 | \( 1 - 1.24iT - 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 11 | \( 1 + 0.520iT - 11T^{2} \) |
| 13 | \( 1 - 1.99iT - 13T^{2} \) |
| 17 | \( 1 + 6.73iT - 17T^{2} \) |
| 19 | \( 1 - 5.18iT - 19T^{2} \) |
| 23 | \( 1 - 8.85T + 23T^{2} \) |
| 29 | \( 1 + 1.05iT - 29T^{2} \) |
| 31 | \( 1 - 0.606T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 + 5.98iT - 47T^{2} \) |
| 53 | \( 1 - 0.483iT - 53T^{2} \) |
| 59 | \( 1 - 4.52T + 59T^{2} \) |
| 61 | \( 1 + 3.69T + 61T^{2} \) |
| 67 | \( 1 - 7.39iT - 67T^{2} \) |
| 71 | \( 1 - 8.64iT - 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 6.96iT - 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 + 16.8iT - 89T^{2} \) |
| 97 | \( 1 - 9.62iT - 97T^{2} \) |
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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
−9.885932103218524353646263608064, −9.248876587890846604104724448616, −8.685533073272917416967384113018, −7.22249196419157555974437265350, −6.59998469022823408885883305572, −5.34820475540252106000708331264, −5.09462918613549100872105579988, −3.75377026760658640790279399553, −2.61599155899892216620146047262, −1.47625272331031982257655071775,
1.17052811048771988771790080473, 2.02715608145207614459568456909, 3.23209565024305951014563346481, 4.65957030201279525023300991812, 5.51892216033876106989181884269, 6.54471836821660517349025842359, 6.92907636403068798582839630671, 8.001693007342693833201062248816, 8.922007098661688348168363515870, 9.700829018943930573190168030511
