L(s) = 1 | + 2-s − 3-s + 4-s − 0.556·5-s − 6-s − 1.46·7-s + 8-s + 9-s − 0.556·10-s − 6.44·11-s − 12-s + 13-s − 1.46·14-s + 0.556·15-s + 16-s − 1.63·17-s + 18-s − 3.33·19-s − 0.556·20-s + 1.46·21-s − 6.44·22-s − 5.42·23-s − 24-s − 4.69·25-s + 26-s − 27-s − 1.46·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.248·5-s − 0.408·6-s − 0.553·7-s + 0.353·8-s + 0.333·9-s − 0.175·10-s − 1.94·11-s − 0.288·12-s + 0.277·13-s − 0.391·14-s + 0.143·15-s + 0.250·16-s − 0.395·17-s + 0.235·18-s − 0.764·19-s − 0.124·20-s + 0.319·21-s − 1.37·22-s − 1.13·23-s − 0.204·24-s − 0.938·25-s + 0.196·26-s − 0.192·27-s − 0.276·28-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
\(L(1)\) |
\(\approx\) |
\(1.165754304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.165754304\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.556T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 11 | \( 1 + 6.44T + 11T^{2} \) |
| 17 | \( 1 + 1.63T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 5.42T + 23T^{2} \) |
| 29 | \( 1 + 9.57T + 29T^{2} \) |
| 31 | \( 1 + 3.33T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 1.49T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 + 5.76T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 6.38T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 0.631T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.69023193212962158420928175672, −7.15514238265048264362568399609, −6.09915855162714601060348861775, −5.80971945901329168252542393257, −5.16828914248288993964290186778, −4.15098696379953776767960091964, −3.82439957826772678784976456414, −2.57588963356262935841457676220, −2.12251153868306706127355921956, −0.45880289925356086404265917917,
0.45880289925356086404265917917, 2.12251153868306706127355921956, 2.57588963356262935841457676220, 3.82439957826772678784976456414, 4.15098696379953776767960091964, 5.16828914248288993964290186778, 5.80971945901329168252542393257, 6.09915855162714601060348861775, 7.15514238265048264362568399609, 7.69023193212962158420928175672