L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 4.46·11-s − 12-s + 5.88·13-s + 16-s − 7.73·17-s − 18-s + 6.61·19-s − 4.46·22-s + 2.61·23-s + 24-s − 5.88·26-s − 27-s − 8.17·29-s + 8.46·31-s − 32-s − 4.46·33-s + 7.73·34-s + 36-s + 3.18·37-s − 6.61·38-s − 5.88·39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.34·11-s − 0.288·12-s + 1.63·13-s + 0.250·16-s − 1.87·17-s − 0.235·18-s + 1.51·19-s − 0.952·22-s + 0.545·23-s + 0.204·24-s − 1.15·26-s − 0.192·27-s − 1.51·29-s + 1.52·31-s − 0.176·32-s − 0.778·33-s + 1.32·34-s + 0.166·36-s + 0.523·37-s − 1.07·38-s − 0.942·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.497137304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.497137304\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4.46T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 - 6.61T + 19T^{2} \) |
| 23 | \( 1 - 2.61T + 23T^{2} \) |
| 29 | \( 1 + 8.17T + 29T^{2} \) |
| 31 | \( 1 - 8.46T + 31T^{2} \) |
| 37 | \( 1 - 3.18T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 - 6.79T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 - 4.66T + 59T^{2} \) |
| 61 | \( 1 - 9.79T + 61T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 - 4.01T + 73T^{2} \) |
| 79 | \( 1 + 6.98T + 79T^{2} \) |
| 83 | \( 1 - 5.35T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 7.71T + 97T^{2} \) |
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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
−7.946784926806489089077710963670, −7.08767621745598171302827757994, −6.55628868468480195409120538062, −6.06103502872205223299896463875, −5.24572993611335713919740492619, −4.17579903772157288950520913244, −3.69151537505228641401331804130, −2.52597035308976948825806439176, −1.42633982284484187661037039915, −0.797061867715446143764064308148,
0.797061867715446143764064308148, 1.42633982284484187661037039915, 2.52597035308976948825806439176, 3.69151537505228641401331804130, 4.17579903772157288950520913244, 5.24572993611335713919740492619, 6.06103502872205223299896463875, 6.55628868468480195409120538062, 7.08767621745598171302827757994, 7.946784926806489089077710963670