| L(s) = 1 | + 1.73i·2-s − 0.999·4-s + 2i·7-s + 1.73i·8-s + 11-s − 5.46i·13-s − 3.46·14-s − 5·16-s − 5.46·19-s + 1.73i·22-s + 6.92i·23-s + 9.46·26-s − 1.99i·28-s − 3.46·29-s − 10.9·31-s − 5.19i·32-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s + 0.755i·7-s + 0.612i·8-s + 0.301·11-s − 1.51i·13-s − 0.925·14-s − 1.25·16-s − 1.25·19-s + 0.369i·22-s + 1.44i·23-s + 1.85·26-s − 0.377i·28-s − 0.643·29-s − 1.96·31-s − 0.918i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6860153426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6860153426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 13 | \( 1 + 5.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92iT - 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 4.92iT - 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 0.928iT - 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 8.39iT - 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 8.53iT - 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.045463865399637524196122466369, −8.624620470353080575662122675259, −7.67017042035867120358407241480, −7.33469465507360509724382511123, −6.21119806104241908709911456345, −5.69490244023560750193203742191, −5.18058364028652628663023411138, −3.97448122760600372419640997376, −2.88145826768297973328942596820, −1.79694761016219889223867217409,
0.20810773747728618761577760139, 1.64349476822250994938637824157, 2.24637538348886894622869036453, 3.56408390987197825658459308347, 4.09241325446444194011138237968, 4.83747016575679997910041884614, 6.32630323042012411979064576657, 6.78828699217701038954719185152, 7.58788386877263929289707635427, 8.953062455311298053551743563124
