| L(s) = 1 | − 0.746i·3-s + (0.782 − 2.09i)5-s + i·7-s + 2.44·9-s − 5.90·11-s + 3.20i·13-s + (−1.56 − 0.584i)15-s − 2.14i·17-s − 3.56·19-s + 0.746·21-s + 3.75i·23-s + (−3.77 − 3.27i)25-s − 4.06i·27-s − 6.61·29-s − 5.79·31-s + ⋯ |
| L(s) = 1 | − 0.431i·3-s + (0.350 − 0.936i)5-s + 0.377i·7-s + 0.814·9-s − 1.78·11-s + 0.890i·13-s + (−0.403 − 0.151i)15-s − 0.521i·17-s − 0.818·19-s + 0.163·21-s + 0.782i·23-s + (−0.754 − 0.655i)25-s − 0.782i·27-s − 1.22·29-s − 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2823503060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2823503060\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.782 + 2.09i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.746iT - 3T^{2} \) |
| 11 | \( 1 + 5.90T + 11T^{2} \) |
| 13 | \( 1 - 3.20iT - 13T^{2} \) |
| 17 | \( 1 + 2.14iT - 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 3.75iT - 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 + 5.79T + 31T^{2} \) |
| 37 | \( 1 + 0.623iT - 37T^{2} \) |
| 41 | \( 1 + 5.43T + 41T^{2} \) |
| 43 | \( 1 + 12.6iT - 43T^{2} \) |
| 47 | \( 1 - 4.31iT - 47T^{2} \) |
| 53 | \( 1 - 2.11iT - 53T^{2} \) |
| 59 | \( 1 + 7.01T + 59T^{2} \) |
| 61 | \( 1 + 1.86T + 61T^{2} \) |
| 67 | \( 1 + 6.88iT - 67T^{2} \) |
| 71 | \( 1 + 1.81T + 71T^{2} \) |
| 73 | \( 1 + 6.11iT - 73T^{2} \) |
| 79 | \( 1 - 4.35T + 79T^{2} \) |
| 83 | \( 1 - 13.1iT - 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 + 9.65iT - 97T^{2} \) |
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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
−8.674399268955481567833069234533, −7.71649559561845180802532918361, −7.27202555634339710497153289233, −6.17122011476938291353381197077, −5.34728243601630313347802963024, −4.77701414539935015817975606291, −3.74066288718324149714555183431, −2.29301202383612851208860604719, −1.69769941474920271424875823702, −0.087493497970473807217410566483,
1.84636606705999207980311595319, 2.84912586012503224045703010444, 3.68014250802647555516235901341, 4.69159090752835442124158267744, 5.51629900099346898789004176550, 6.31948926001850256246640147268, 7.29842736880156237154398455962, 7.75370114138566519043443208847, 8.642327713093344748941721403865, 9.845162315825135807162335351961
