L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 0.487·7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 6.29·13-s + 0.487·14-s + 15-s + 16-s + 17-s + 18-s + 6.12·19-s − 20-s − 0.487·21-s + 22-s + 5.67·23-s − 24-s + 25-s + 6.29·26-s − 27-s + 0.487·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 0.184·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.74·13-s + 0.130·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.40·19-s − 0.223·20-s − 0.106·21-s + 0.213·22-s + 1.18·23-s − 0.204·24-s + 0.200·25-s + 1.23·26-s − 0.192·27-s + 0.0921·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.941849031\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.941849031\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 0.487T + 7T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 19 | \( 1 - 6.12T + 19T^{2} \) |
| 23 | \( 1 - 5.67T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 0.487T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 - 0.165T + 41T^{2} \) |
| 43 | \( 1 + 8.77T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 0.165T + 53T^{2} \) |
| 59 | \( 1 - 9.22T + 59T^{2} \) |
| 61 | \( 1 - 1.31T + 61T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 - 4.97T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 - 17.6T + 89T^{2} \) |
| 97 | \( 1 + 0.652T + 97T^{2} \) |
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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
−7.985421924231678470779949388565, −7.21323086523635297606104203658, −6.71417305210205820158755781391, −5.74228178120131068548733725775, −5.40379662218185805188626152747, −4.49627693943054303225112968692, −3.61913498967279245212586834267, −3.24650671018977642333778941186, −1.74029295483515154483849240480, −0.912458312929248341552484816906,
0.912458312929248341552484816906, 1.74029295483515154483849240480, 3.24650671018977642333778941186, 3.61913498967279245212586834267, 4.49627693943054303225112968692, 5.40379662218185805188626152747, 5.74228178120131068548733725775, 6.71417305210205820158755781391, 7.21323086523635297606104203658, 7.985421924231678470779949388565