| L(s) = 1 | + 2-s − 3-s + 4-s + 3.18·5-s − 6-s + 0.508·7-s + 8-s + 9-s + 3.18·10-s + 11-s − 12-s + 0.508·14-s − 3.18·15-s + 16-s − 1.18·17-s + 18-s + 19-s + 3.18·20-s − 0.508·21-s + 22-s + 7.36·23-s − 24-s + 5.17·25-s − 27-s + 0.508·28-s − 9.53·29-s − 3.18·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.42·5-s − 0.408·6-s + 0.192·7-s + 0.353·8-s + 0.333·9-s + 1.00·10-s + 0.301·11-s − 0.288·12-s + 0.135·14-s − 0.823·15-s + 0.250·16-s − 0.288·17-s + 0.235·18-s + 0.229·19-s + 0.713·20-s − 0.110·21-s + 0.213·22-s + 1.53·23-s − 0.204·24-s + 1.03·25-s − 0.192·27-s + 0.0960·28-s − 1.77·29-s − 0.582·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.908799160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908799160\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 - 0.508T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 23 | \( 1 - 7.36T + 23T^{2} \) |
| 29 | \( 1 + 9.53T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 37 | \( 1 - 4.04T + 37T^{2} \) |
| 41 | \( 1 + 9.74T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 0.129T + 53T^{2} \) |
| 59 | \( 1 + 7.74T + 59T^{2} \) |
| 61 | \( 1 + 7.06T + 61T^{2} \) |
| 67 | \( 1 - 9.56T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.81T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.706204471945465236945291105467, −9.169857456620790870356461027297, −7.925036379224523473736107037620, −6.75764116923399700879048016389, −6.35639502830922980892388477456, −5.32136143288378522937095158200, −4.94286532816259896644578094023, −3.61006116920318506759304408828, −2.37845574161286393790094045573, −1.34864706644503212344032379864,
1.34864706644503212344032379864, 2.37845574161286393790094045573, 3.61006116920318506759304408828, 4.94286532816259896644578094023, 5.32136143288378522937095158200, 6.35639502830922980892388477456, 6.75764116923399700879048016389, 7.925036379224523473736107037620, 9.169857456620790870356461027297, 9.706204471945465236945291105467
