| L(s) = 1 | + 2.67·2-s + 5.15·4-s − 2.80·7-s + 8.44·8-s + 11-s + 5.11·13-s − 7.50·14-s + 12.2·16-s + 4.54·17-s − 4.57·19-s + 2.67·22-s + 4·23-s + 13.6·26-s − 14.4·28-s + 2.38·29-s − 0.962·31-s + 15.9·32-s + 12.1·34-s − 1.61·37-s − 12.2·38-s + 2.38·41-s − 2.80·43-s + 5.15·44-s + 10.7·46-s − 4.31·47-s + 0.873·49-s + 26.3·52-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.57·4-s − 1.06·7-s + 2.98·8-s + 0.301·11-s + 1.41·13-s − 2.00·14-s + 3.06·16-s + 1.10·17-s − 1.04·19-s + 0.570·22-s + 0.834·23-s + 2.68·26-s − 2.73·28-s + 0.443·29-s − 0.172·31-s + 2.81·32-s + 2.08·34-s − 0.265·37-s − 1.98·38-s + 0.372·41-s − 0.427·43-s + 0.777·44-s + 1.57·46-s − 0.629·47-s + 0.124·49-s + 3.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.141399916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.141399916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 7 | \( 1 + 2.80T + 7T^{2} \) |
| 13 | \( 1 - 5.11T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 19 | \( 1 + 4.57T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 + 0.962T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 - 2.38T + 41T^{2} \) |
| 43 | \( 1 + 2.80T + 43T^{2} \) |
| 47 | \( 1 + 4.31T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 7.92T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 - 7.35T + 71T^{2} \) |
| 73 | \( 1 + 6.41T + 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 + 0.806T + 83T^{2} \) |
| 89 | \( 1 - 2.96T + 89T^{2} \) |
| 97 | \( 1 - 9.92T + 97T^{2} \) |
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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
−8.857846498097446685269635268276, −7.921750100089480391720263638626, −6.84540100021831471455704906482, −6.43844244304904878367628584796, −5.79306765203612391501947305796, −4.98202639714813758689288300376, −3.92729288341920008085240274453, −3.47997674795130572765919602868, −2.66465030038503452559749227258, −1.36397059200448912322557757740,
1.36397059200448912322557757740, 2.66465030038503452559749227258, 3.47997674795130572765919602868, 3.92729288341920008085240274453, 4.98202639714813758689288300376, 5.79306765203612391501947305796, 6.43844244304904878367628584796, 6.84540100021831471455704906482, 7.921750100089480391720263638626, 8.857846498097446685269635268276
