| L(s) = 1 | + 3-s − 1.90i·5-s − 4.43i·7-s + 9-s − i·11-s + (2.32 − 2.75i)13-s − 1.90i·15-s − 1.12·17-s − 6.14i·19-s − 4.43i·21-s + 4.17·23-s + 1.35·25-s + 27-s + 0.694·29-s + 0.657i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.853i·5-s − 1.67i·7-s + 0.333·9-s − 0.301i·11-s + (0.644 − 0.764i)13-s − 0.492i·15-s − 0.273·17-s − 1.40i·19-s − 0.968i·21-s + 0.870·23-s + 0.271·25-s + 0.192·27-s + 0.129·29-s + 0.118i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.287903724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.287903724\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (-2.32 + 2.75i)T \) |
| good | 5 | \( 1 + 1.90iT - 5T^{2} \) |
| 7 | \( 1 + 4.43iT - 7T^{2} \) |
| 17 | \( 1 + 1.12T + 17T^{2} \) |
| 19 | \( 1 + 6.14iT - 19T^{2} \) |
| 23 | \( 1 - 4.17T + 23T^{2} \) |
| 29 | \( 1 - 0.694T + 29T^{2} \) |
| 31 | \( 1 - 0.657iT - 31T^{2} \) |
| 37 | \( 1 - 6.76iT - 37T^{2} \) |
| 41 | \( 1 - 5.69iT - 41T^{2} \) |
| 43 | \( 1 + 1.69T + 43T^{2} \) |
| 47 | \( 1 - 1.70iT - 47T^{2} \) |
| 53 | \( 1 + 3.66T + 53T^{2} \) |
| 59 | \( 1 - 5.09iT - 59T^{2} \) |
| 61 | \( 1 - 0.988T + 61T^{2} \) |
| 67 | \( 1 + 8.43iT - 67T^{2} \) |
| 71 | \( 1 - 13.7iT - 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 + 14.3iT - 83T^{2} \) |
| 89 | \( 1 + 1.93iT - 89T^{2} \) |
| 97 | \( 1 - 9.00iT - 97T^{2} \) |
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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
−8.395596185042354023478504838973, −7.64134889159625344174150892800, −6.97967410901689463374305185780, −6.23389213103838789881706390138, −4.90384258816145953957561529307, −4.60281777822764885701594056768, −3.56311340702230281385467907291, −2.88715096172106514525444657765, −1.31878872990101505637627397274, −0.67658066817852354820312858245,
1.70628621449781434295205458531, 2.41218638670690944420589358437, 3.22451050436278810505632477722, 4.04700186529150421054509684583, 5.17820341165705938463342638138, 5.94633017516194497564856647248, 6.62973996452139189042112870051, 7.36001826939317765210384093125, 8.319727528461275380134991740819, 8.817594059977533203128696787409
