| L(s) = 1 | + 1.41i·3-s + 2.82i·5-s + 0.999·9-s + 1.41i·11-s + 4·13-s − 4.00·15-s + (−3 − 2.82i)17-s + 4·19-s + 1.41i·23-s − 3.00·25-s + 5.65i·27-s − 2.82i·29-s + 4.24i·31-s − 2.00·33-s + 8.48i·37-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s + 1.26i·5-s + 0.333·9-s + 0.426i·11-s + 1.10·13-s − 1.03·15-s + (−0.727 − 0.685i)17-s + 0.917·19-s + 0.294i·23-s − 0.600·25-s + 1.08i·27-s − 0.525i·29-s + 0.762i·31-s − 0.348·33-s + 1.39i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.018405627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018405627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (3 + 2.82i)T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 4.24iT - 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 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
−9.171924702527574337555717665426, −8.117150631464355883057345796486, −7.30460009731033303507926265645, −6.71585799606762800398881691780, −5.99073947578360479849892432441, −4.94533604186819072799102090721, −4.23651806592263843561796399330, −3.35557915310758406773997657842, −2.73350639811331207734201520511, −1.37450596698300350519268658808,
0.69197666760147965336248626507, 1.39188626362306616951778318912, 2.40401879973578705823224118485, 3.84723019786387171980301979499, 4.33556329033103752053931528617, 5.55971352873620900131879208340, 5.92416336827612259671591408493, 6.98330175918415174496554911145, 7.58244010693781146472357456071, 8.478853418615852924070730515768
