| L(s) = 1 | + 2-s + 3-s + 4-s + 4.08·5-s + 6-s + 7-s + 8-s + 9-s + 4.08·10-s + 11-s + 12-s + 13-s + 14-s + 4.08·15-s + 16-s − 4.36·17-s + 18-s + 1.18·19-s + 4.08·20-s + 21-s + 22-s + 2.13·23-s + 24-s + 11.6·25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.82·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.29·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 1.05·15-s + 0.250·16-s − 1.05·17-s + 0.235·18-s + 0.271·19-s + 0.913·20-s + 0.218·21-s + 0.213·22-s + 0.445·23-s + 0.204·24-s + 2.33·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.346140060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.346140060\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 4.08T + 5T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 1.18T + 19T^{2} \) |
| 23 | \( 1 - 2.13T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 4.22T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 + 9.15T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 8.56T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 4.04T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 0.644T + 83T^{2} \) |
| 89 | \( 1 + 2.31T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.123963761973419288247074003646, −7.13906262639398638972172404492, −6.49427520410639089079332960652, −5.99992239169041826479570250412, −5.10727329747081917441415730642, −4.64371528162391770948827330701, −3.54817762301822233828105222704, −2.70553972737776630225680417523, −1.98480801380289635093308787923, −1.33485406214221930214273470003,
1.33485406214221930214273470003, 1.98480801380289635093308787923, 2.70553972737776630225680417523, 3.54817762301822233828105222704, 4.64371528162391770948827330701, 5.10727329747081917441415730642, 5.99992239169041826479570250412, 6.49427520410639089079332960652, 7.13906262639398638972172404492, 8.123963761973419288247074003646
