| L(s) = 1 | + 3-s + i·5-s − 0.561i·7-s + 9-s − 2.56i·11-s + (3.56 + 0.561i)13-s + i·15-s + 4.56·17-s − 3.12i·19-s − 0.561i·21-s − 2.56·23-s − 25-s + 27-s − 2·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447i·5-s − 0.212i·7-s + 0.333·9-s − 0.772i·11-s + (0.987 + 0.155i)13-s + 0.258i·15-s + 1.10·17-s − 0.716i·19-s − 0.122i·21-s − 0.534·23-s − 0.200·25-s + 0.192·27-s − 0.371·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260354762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260354762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-3.56 - 0.561i)T \) |
| good | 7 | \( 1 + 0.561iT - 7T^{2} \) |
| 11 | \( 1 + 2.56iT - 11T^{2} \) |
| 17 | \( 1 - 4.56T + 17T^{2} \) |
| 19 | \( 1 + 3.12iT - 19T^{2} \) |
| 23 | \( 1 + 2.56T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 - 1.43iT - 37T^{2} \) |
| 41 | \( 1 + 5.68iT - 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 - 10.2iT - 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 + 9.36iT - 67T^{2} \) |
| 71 | \( 1 + 5.43iT - 71T^{2} \) |
| 73 | \( 1 - 9.12iT - 73T^{2} \) |
| 79 | \( 1 - 3.68T + 79T^{2} \) |
| 83 | \( 1 + 5.12iT - 83T^{2} \) |
| 89 | \( 1 - 4.56iT - 89T^{2} \) |
| 97 | \( 1 + 2.56iT - 97T^{2} \) |
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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
−9.268922052228342259344893963105, −8.667302676797166024800687122601, −7.83527024934782872373596927476, −7.12899607361731709700934418837, −6.17514903232155148876978524905, −5.42388169299074363350989532818, −4.06618285793661004751734627016, −3.43460018745110648319519130188, −2.44039755837552907615156142815, −1.03117964914297343101637268739,
1.20585832586164213704937753322, 2.30508140081603299010927607797, 3.54739189761823707952537163069, 4.24191734411024862170000301152, 5.40906282116556334276272930331, 6.09563594177782928929689385451, 7.25807193045122496517272603936, 7.970446879081797592781873438295, 8.587605900111751774025351639427, 9.488829431378352702321983265395
