| L(s) = 1 | − 2-s − 3-s + 4-s + 1.64·5-s + 6-s − 8-s + 9-s − 1.64·10-s + 2·11-s − 12-s − 5.64·13-s − 1.64·15-s + 16-s − 7.29·17-s − 18-s − 19-s + 1.64·20-s − 2·22-s − 3.64·23-s + 24-s − 2.29·25-s + 5.64·26-s − 27-s + 9.29·29-s + 1.64·30-s − 1.64·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.736·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.520·10-s + 0.603·11-s − 0.288·12-s − 1.56·13-s − 0.424·15-s + 0.250·16-s − 1.76·17-s − 0.235·18-s − 0.229·19-s + 0.368·20-s − 0.426·22-s − 0.760·23-s + 0.204·24-s − 0.458·25-s + 1.10·26-s − 0.192·27-s + 1.72·29-s + 0.300·30-s − 0.295·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8679826003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8679826003\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 9.29T + 29T^{2} \) |
| 31 | \( 1 + 1.64T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 9.29T + 43T^{2} \) |
| 47 | \( 1 - 1.64T + 47T^{2} \) |
| 53 | \( 1 - 9.29T + 53T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 - 9.29T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.278322139362857828925034421953, −7.30325518339599675671511298379, −6.60493282471615429147740220139, −6.35638034737062026471441875136, −5.23215853473317155055103623043, −4.71379528356334513084119585637, −3.68130523680429217727857384303, −2.30041002272649372235753665788, −1.98331554899864010739635114373, −0.54539230259678095305437303916,
0.54539230259678095305437303916, 1.98331554899864010739635114373, 2.30041002272649372235753665788, 3.68130523680429217727857384303, 4.71379528356334513084119585637, 5.23215853473317155055103623043, 6.35638034737062026471441875136, 6.60493282471615429147740220139, 7.30325518339599675671511298379, 8.278322139362857828925034421953
