| L(s) = 1 | + 1.93·2-s + (−1.41 − i)3-s + 1.73·4-s − 1.41i·5-s + (−2.73 − 1.93i)6-s − 3.73·7-s − 0.517·8-s + (1.00 + 2.82i)9-s − 2.73i·10-s − 0.378i·11-s + (−2.44 − 1.73i)12-s + 3.73i·13-s − 7.20·14-s + (−1.41 + 2.00i)15-s − 4.46·16-s + 4.89i·17-s + ⋯ |
| L(s) = 1 | + 1.36·2-s + (−0.816 − 0.577i)3-s + 0.866·4-s − 0.632i·5-s + (−1.11 − 0.788i)6-s − 1.41·7-s − 0.183·8-s + (0.333 + 0.942i)9-s − 0.863i·10-s − 0.114i·11-s + (−0.707 − 0.500i)12-s + 1.03i·13-s − 1.92·14-s + (−0.365 + 0.516i)15-s − 1.11·16-s + 1.18i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.519 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4478636743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4478636743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.41 + i)T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.93T + 2T^{2} \) |
| 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 3.73T + 7T^{2} \) |
| 11 | \( 1 + 0.378iT - 11T^{2} \) |
| 13 | \( 1 - 3.73iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 23 | \( 1 + 0.378iT - 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 - 4.46iT - 31T^{2} \) |
| 37 | \( 1 - 4.26iT - 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 + 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 6.03T + 53T^{2} \) |
| 59 | \( 1 + 8.38T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 3.58T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 - 3.53iT - 79T^{2} \) |
| 83 | \( 1 + 7.72iT - 83T^{2} \) |
| 89 | \( 1 + 7.34T + 89T^{2} \) |
| 97 | \( 1 + 7.46iT - 97T^{2} \) |
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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
−10.35218256932280044206653215323, −9.338538430499265020308239751338, −8.530745506799453455112190204398, −7.12112144266719278742821091063, −6.48480619956398438016198034463, −5.89583535278803837012264792761, −5.03223510332054760891771701670, −4.15308511941986395151917755355, −3.21282983175359208204945121368, −1.76646011912816472018857575109,
0.13169294075568819825756914978, 2.86605895112245909088193523845, 3.35753634441265187027336525478, 4.34496318183553996138189047727, 5.34448190432744563937826393463, 5.97187234487208155233594597899, 6.66539047479683833599349528636, 7.45524685906077741138101221605, 9.237322309792585572173386852726, 9.636279302848096912208739119401
