L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 4.28·7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 1.90·13-s + 4.28·14-s − 15-s + 16-s − 17-s + 18-s − 1.90·19-s + 20-s − 4.28·21-s + 22-s + 7.75·23-s − 24-s + 25-s − 1.90·26-s − 27-s + 4.28·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 1.61·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.529·13-s + 1.14·14-s − 0.258·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 0.437·19-s + 0.223·20-s − 0.934·21-s + 0.213·22-s + 1.61·23-s − 0.204·24-s + 0.200·25-s − 0.374·26-s − 0.192·27-s + 0.809·28-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
\(L(1)\) |
\(\approx\) |
\(3.725610721\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.725610721\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 4.28T + 7T^{2} \) |
| 13 | \( 1 + 1.90T + 13T^{2} \) |
| 19 | \( 1 + 1.90T + 19T^{2} \) |
| 23 | \( 1 - 7.75T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 1.47T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 - 7.95T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 - 0.228T + 73T^{2} \) |
| 79 | \( 1 + 7.13T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 1.03T + 97T^{2} \) |
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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
−8.103209463035834529028353130937, −7.08464189039199712054557986508, −6.83116092556924905076751927573, −5.67872714906346975995128654962, −5.25700323691236746620961134368, −4.63188180417710058311753237506, −3.98370257575816272435485575757, −2.71714739082794751497614994099, −1.89780511604782510983952109183, −1.02389634848355937172787332811,
1.02389634848355937172787332811, 1.89780511604782510983952109183, 2.71714739082794751497614994099, 3.98370257575816272435485575757, 4.63188180417710058311753237506, 5.25700323691236746620961134368, 5.67872714906346975995128654962, 6.83116092556924905076751927573, 7.08464189039199712054557986508, 8.103209463035834529028353130937