| L(s) = 1 | − 1.78·3-s + (−0.615 + 2.14i)5-s + 2.64·7-s + 0.171·9-s + 0.737i·13-s + (1.09 − 3.82i)15-s + 5.43i·19-s − 4.71·21-s + 7.48·23-s + (−4.24 − 2.64i)25-s + 5.03·27-s + (−1.62 + 5.68i)35-s − 1.31i·39-s + (−0.105 + 0.368i)45-s + 7.00·49-s + ⋯ |
| L(s) = 1 | − 1.02·3-s + (−0.275 + 0.961i)5-s + 0.999·7-s + 0.0571·9-s + 0.204i·13-s + (0.282 − 0.988i)15-s + 1.24i·19-s − 1.02·21-s + 1.56·23-s + (−0.848 − 0.529i)25-s + 0.969·27-s + (−0.275 + 0.961i)35-s − 0.210i·39-s + (−0.0157 + 0.0549i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.485 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9616288835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9616288835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.615 - 2.14i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 3 | \( 1 + 1.78T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.737iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.43iT - 19T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.45iT - 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.65iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 16.9iT - 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.326285960726779521860254303791, −8.376813950716308741138448922249, −7.65280209671272463793348924265, −6.89841886538176702705384848708, −6.14114196812719016939424437475, −5.39969805986111196531644123061, −4.62545245249775395333343741584, −3.61476798214318105586844082452, −2.52710271925341926925361255371, −1.22634152846036850955283407651,
0.44425301532945436533360114095, 1.42702263101298294303620256823, 2.86636679804492277747508363113, 4.25014548267609177094248149290, 5.04496857948754845057083667189, 5.26742611516109744918470256419, 6.34556284191767889400955022979, 7.23589582219836689748840316287, 8.041682424565355435799448677098, 8.819062136631608483791139401382
