L(s) = 1 | − 2-s − 3-s + 4-s − 1.82·5-s + 6-s + 1.45·7-s − 8-s + 9-s + 1.82·10-s − 0.513·11-s − 12-s − 13-s − 1.45·14-s + 1.82·15-s + 16-s − 0.785·17-s − 18-s − 1.12·19-s − 1.82·20-s − 1.45·21-s + 0.513·22-s + 5.13·23-s + 24-s − 1.65·25-s + 26-s − 27-s + 1.45·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.818·5-s + 0.408·6-s + 0.550·7-s − 0.353·8-s + 0.333·9-s + 0.578·10-s − 0.154·11-s − 0.288·12-s − 0.277·13-s − 0.389·14-s + 0.472·15-s + 0.250·16-s − 0.190·17-s − 0.235·18-s − 0.258·19-s − 0.409·20-s − 0.317·21-s + 0.109·22-s + 1.07·23-s + 0.204·24-s − 0.330·25-s + 0.196·26-s − 0.192·27-s + 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 1.82T + 5T^{2} \) |
| 7 | \( 1 - 1.45T + 7T^{2} \) |
| 11 | \( 1 + 0.513T + 11T^{2} \) |
| 17 | \( 1 + 0.785T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 5.13T + 23T^{2} \) |
| 29 | \( 1 - 3.60T + 29T^{2} \) |
| 31 | \( 1 + 8.18T + 31T^{2} \) |
| 37 | \( 1 - 2.66T + 37T^{2} \) |
| 41 | \( 1 - 5.68T + 41T^{2} \) |
| 43 | \( 1 - 5.80T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 + 9.91T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 9.58T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 - 4.34T + 71T^{2} \) |
| 73 | \( 1 - 0.434T + 73T^{2} \) |
| 79 | \( 1 - 7.70T + 79T^{2} \) |
| 83 | \( 1 + 0.451T + 83T^{2} \) |
| 89 | \( 1 - 0.196T + 89T^{2} \) |
| 97 | \( 1 - 0.774T + 97T^{2} \) |
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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
−7.73891564183470643958565446483, −6.94793508024894316360164286658, −6.22912580033590146966260361381, −5.41620032777262868491908973763, −4.68552033039490786556196143063, −3.98873607318810571579645449218, −3.01798845671054274716053972263, −2.03888111209899837568691444773, −1.01658866575865873078021936617, 0,
1.01658866575865873078021936617, 2.03888111209899837568691444773, 3.01798845671054274716053972263, 3.98873607318810571579645449218, 4.68552033039490786556196143063, 5.41620032777262868491908973763, 6.22912580033590146966260361381, 6.94793508024894316360164286658, 7.73891564183470643958565446483