| L(s) = 1 | − i·3-s + 4i·7-s + 2·9-s + 3·11-s + 2i·13-s − i·17-s + 19-s + 4·21-s − i·23-s − 5i·27-s + 8·31-s − 3i·33-s − 2i·37-s + 2·39-s + 41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.51i·7-s + 0.666·9-s + 0.904·11-s + 0.554i·13-s − 0.242i·17-s + 0.229·19-s + 0.872·21-s − 0.208i·23-s − 0.962i·27-s + 1.43·31-s − 0.522i·33-s − 0.328i·37-s + 0.320·39-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.275233371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.275233371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - T + 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 17iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 5iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.437337379839484371540971659192, −7.62170760911705284933810906396, −6.80898046753625063647663753487, −6.31654083120569311513666372809, −5.56441036576916089129040300008, −4.67144239365337661307511003678, −3.87941175318656998710595046459, −2.71458336825230376016305629009, −2.02301267821759488656506210604, −1.05391247162647877567154929239,
0.77239985569542837796107523627, 1.61358980217112183307283460313, 3.16444861373754063235514468953, 3.78435176414386032732517711557, 4.48007040144183224538160456564, 5.02420194153362776616006040126, 6.33803540962975711997588809020, 6.70333632315593542592872579336, 7.67860906691481902844870758112, 8.035844255056520598016159852902
