| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 2.86·7-s − 8-s + 9-s − 10-s + 12-s + 5.25·13-s + 2.86·14-s + 15-s + 16-s − 0.154·17-s − 18-s − 1.46·19-s + 20-s − 2.86·21-s + 3.78·23-s − 24-s + 25-s − 5.25·26-s + 27-s − 2.86·28-s + 3.48·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 1.08·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.288·12-s + 1.45·13-s + 0.766·14-s + 0.258·15-s + 0.250·16-s − 0.0375·17-s − 0.235·18-s − 0.335·19-s + 0.223·20-s − 0.625·21-s + 0.789·23-s − 0.204·24-s + 0.200·25-s − 1.03·26-s + 0.192·27-s − 0.542·28-s + 0.647·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735497938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735497938\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 2.86T + 7T^{2} \) |
| 13 | \( 1 - 5.25T + 13T^{2} \) |
| 17 | \( 1 + 0.154T + 17T^{2} \) |
| 19 | \( 1 + 1.46T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 - 3.48T + 29T^{2} \) |
| 31 | \( 1 - 2.98T + 31T^{2} \) |
| 37 | \( 1 - 8.85T + 37T^{2} \) |
| 41 | \( 1 + 8.41T + 41T^{2} \) |
| 43 | \( 1 + 7.86T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 + 1.73T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.54T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 6.78T + 79T^{2} \) |
| 83 | \( 1 - 1.52T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
| 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
−8.573767169794892546027912730576, −8.108057715286497544140553667465, −6.95135047613777656163255710337, −6.51225928654338508632666538820, −5.86240335268329111813354578157, −4.68978772083149847956054285484, −3.52160548744005596348362890616, −3.01559758226567711089938181440, −1.93274471037415795064004195451, −0.849797425940650697161836309328,
0.849797425940650697161836309328, 1.93274471037415795064004195451, 3.01559758226567711089938181440, 3.52160548744005596348362890616, 4.68978772083149847956054285484, 5.86240335268329111813354578157, 6.51225928654338508632666538820, 6.95135047613777656163255710337, 8.108057715286497544140553667465, 8.573767169794892546027912730576
