L(s) = 1 | + 2-s + 3-s + 4-s − 0.111·5-s + 6-s − 0.538·7-s + 8-s + 9-s − 0.111·10-s + 1.57·11-s + 12-s + 13-s − 0.538·14-s − 0.111·15-s + 16-s − 6.71·17-s + 18-s − 1.53·19-s − 0.111·20-s − 0.538·21-s + 1.57·22-s + 3.09·23-s + 24-s − 4.98·25-s + 26-s + 27-s − 0.538·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.0499·5-s + 0.408·6-s − 0.203·7-s + 0.353·8-s + 0.333·9-s − 0.0353·10-s + 0.474·11-s + 0.288·12-s + 0.277·13-s − 0.143·14-s − 0.0288·15-s + 0.250·16-s − 1.62·17-s + 0.235·18-s − 0.352·19-s − 0.0249·20-s − 0.117·21-s + 0.335·22-s + 0.644·23-s + 0.204·24-s − 0.997·25-s + 0.196·26-s + 0.192·27-s − 0.101·28-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
\(L(1)\) |
\(\approx\) |
\(4.178334971\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.178334971\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.111T + 5T^{2} \) |
| 7 | \( 1 + 0.538T + 7T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 17 | \( 1 + 6.71T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 - 3.09T + 23T^{2} \) |
| 29 | \( 1 - 8.44T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 - 7.67T + 43T^{2} \) |
| 47 | \( 1 - 0.691T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 4.99T + 67T^{2} \) |
| 71 | \( 1 - 7.45T + 71T^{2} \) |
| 73 | \( 1 - 3.57T + 73T^{2} \) |
| 79 | \( 1 - 9.41T + 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 + 6.10T + 97T^{2} \) |
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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
−7.83747077590268173299863133259, −6.93273774926932404295901925989, −6.49499791207436684366139022249, −5.84966263094737003219993337975, −4.76813525227839165569975441125, −4.27672512705784176928312855720, −3.61615855747056360315323439852, −2.66268821206349287379230103051, −2.11946970445257171129143412609, −0.888228473045301792293649017106,
0.888228473045301792293649017106, 2.11946970445257171129143412609, 2.66268821206349287379230103051, 3.61615855747056360315323439852, 4.27672512705784176928312855720, 4.76813525227839165569975441125, 5.84966263094737003219993337975, 6.49499791207436684366139022249, 6.93273774926932404295901925989, 7.83747077590268173299863133259