| L(s) = 1 | − 2-s − 3-s + 4-s − 0.554·5-s + 6-s + 7-s − 8-s + 9-s + 0.554·10-s + 0.356·11-s − 12-s − 14-s + 0.554·15-s + 16-s − 3.80·17-s − 18-s − 5.74·19-s − 0.554·20-s − 21-s − 0.356·22-s − 2.44·23-s + 24-s − 4.69·25-s − 27-s + 28-s + 0.405·29-s − 0.554·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.248·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.175·10-s + 0.107·11-s − 0.288·12-s − 0.267·14-s + 0.143·15-s + 0.250·16-s − 0.922·17-s − 0.235·18-s − 1.31·19-s − 0.124·20-s − 0.218·21-s − 0.0760·22-s − 0.509·23-s + 0.204·24-s − 0.938·25-s − 0.192·27-s + 0.188·28-s + 0.0753·29-s − 0.101·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7002460633\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7002460633\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.554T + 5T^{2} \) |
| 11 | \( 1 - 0.356T + 11T^{2} \) |
| 17 | \( 1 + 3.80T + 17T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 + 2.44T + 23T^{2} \) |
| 29 | \( 1 - 0.405T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 3.54T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 + 0.603T + 47T^{2} \) |
| 53 | \( 1 + 1.35T + 53T^{2} \) |
| 59 | \( 1 + 5.54T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 7.42T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 4.80T + 89T^{2} \) |
| 97 | \( 1 + 2.65T + 97T^{2} \) |
| 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
−8.085661365533225188939820151009, −7.25559270490540527029359766238, −6.51850790934860162605601869789, −6.10895790391267112897609575707, −5.11188964812853834471134234842, −4.38468957820710088999350848869, −3.66011137781667334840890740850, −2.39167573010757554282269512651, −1.73108018270813567346492154628, −0.47883527559379491527470421676,
0.47883527559379491527470421676, 1.73108018270813567346492154628, 2.39167573010757554282269512651, 3.66011137781667334840890740850, 4.38468957820710088999350848869, 5.11188964812853834471134234842, 6.10895790391267112897609575707, 6.51850790934860162605601869789, 7.25559270490540527029359766238, 8.085661365533225188939820151009
