| L(s) = 1 | + 1.44·3-s + i·5-s + (−2.48 − 0.919i)7-s − 0.912·9-s − 5.66i·11-s + 5.14i·13-s + 1.44i·15-s + 1.65i·17-s − 1.33·19-s + (−3.58 − 1.32i)21-s + 3.62i·23-s − 25-s − 5.65·27-s + 0.0482·29-s + 3.08·31-s + ⋯ |
| L(s) = 1 | + 0.834·3-s + 0.447i·5-s + (−0.937 − 0.347i)7-s − 0.304·9-s − 1.70i·11-s + 1.42i·13-s + 0.373i·15-s + 0.401i·17-s − 0.307·19-s + (−0.782 − 0.289i)21-s + 0.755i·23-s − 0.200·25-s − 1.08·27-s + 0.00895·29-s + 0.554·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4078430515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4078430515\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (2.48 + 0.919i)T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 11 | \( 1 + 5.66iT - 11T^{2} \) |
| 13 | \( 1 - 5.14iT - 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 3.62iT - 23T^{2} \) |
| 29 | \( 1 - 0.0482T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 2.37iT - 41T^{2} \) |
| 43 | \( 1 - 6.18iT - 43T^{2} \) |
| 47 | \( 1 + 7.75T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 2.30iT - 61T^{2} \) |
| 67 | \( 1 + 0.207iT - 67T^{2} \) |
| 71 | \( 1 - 6.67iT - 71T^{2} \) |
| 73 | \( 1 - 3.42iT - 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 4.28iT - 89T^{2} \) |
| 97 | \( 1 - 12.1iT - 97T^{2} \) |
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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
−9.316647638672020088772498392260, −8.589795885168437292641262593946, −8.047306812265671558908988854371, −6.95990695185256499921672269848, −6.36729102673704164741582391556, −5.65500461804438281205583268682, −4.26250935875966184738391561008, −3.35075158866821069397236773918, −3.00925254617546738176518663894, −1.67757576557642177017544034555,
0.11405959280656554460455777061, 1.92563805449427215429308865066, 2.80705517028080280194583349905, 3.54657500245365749486917865714, 4.69189673163906176241082036903, 5.43759611513058733082755502803, 6.42712522935836234988103394255, 7.26398066880770258206904509288, 8.035350570114270320699816231698, 8.727318405209517418042646988372
