L(s) = 1 | − 2-s + 3-s + 4-s − 3.56·5-s − 6-s − 0.117·7-s − 8-s + 9-s + 3.56·10-s + 6.11·11-s + 12-s + 3.74·13-s + 0.117·14-s − 3.56·15-s + 16-s − 17-s − 18-s − 0.663·19-s − 3.56·20-s − 0.117·21-s − 6.11·22-s + 8.53·23-s − 24-s + 7.71·25-s − 3.74·26-s + 27-s − 0.117·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.59·5-s − 0.408·6-s − 0.0445·7-s − 0.353·8-s + 0.333·9-s + 1.12·10-s + 1.84·11-s + 0.288·12-s + 1.03·13-s + 0.0315·14-s − 0.920·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.152·19-s − 0.797·20-s − 0.0257·21-s − 1.30·22-s + 1.77·23-s − 0.204·24-s + 1.54·25-s − 0.735·26-s + 0.192·27-s − 0.0222·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.578848069\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.578848069\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 + 3.56T + 5T^{2} \) |
| 7 | \( 1 + 0.117T + 7T^{2} \) |
| 11 | \( 1 - 6.11T + 11T^{2} \) |
| 13 | \( 1 - 3.74T + 13T^{2} \) |
| 19 | \( 1 + 0.663T + 19T^{2} \) |
| 23 | \( 1 - 8.53T + 23T^{2} \) |
| 29 | \( 1 - 7.62T + 29T^{2} \) |
| 31 | \( 1 + 7.73T + 31T^{2} \) |
| 37 | \( 1 + 2.71T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 - 0.835T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 - 5.50T + 53T^{2} \) |
| 61 | \( 1 - 5.26T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 3.38T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 - 0.152T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 3.96T + 89T^{2} \) |
| 97 | \( 1 - 2.65T + 97T^{2} \) |
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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
−8.287946207678469844809980971566, −7.40209041030776685424658012727, −6.86322223639197837620675803938, −6.40216941415759868550133942353, −5.06777725509253996427669523120, −4.07053221144089297426757594710, −3.66803928775259819632959632962, −2.94736205863431346984677098465, −1.55336074914897799457765500156, −0.77731502209312401205209944972,
0.77731502209312401205209944972, 1.55336074914897799457765500156, 2.94736205863431346984677098465, 3.66803928775259819632959632962, 4.07053221144089297426757594710, 5.06777725509253996427669523120, 6.40216941415759868550133942353, 6.86322223639197837620675803938, 7.40209041030776685424658012727, 8.287946207678469844809980971566