| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 17.5·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 35.0·10-s − 3.08·11-s − 12·12-s + 13·13-s − 14·14-s − 52.5·15-s + 16·16-s + 124.·17-s + 18·18-s − 78.9·19-s + 70.0·20-s + 21·21-s − 6.16·22-s + 126.·23-s − 24·24-s + 181.·25-s + 26·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.56·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.10·10-s − 0.0844·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.903·15-s + 0.250·16-s + 1.77·17-s + 0.235·18-s − 0.953·19-s + 0.782·20-s + 0.218·21-s − 0.0597·22-s + 1.14·23-s − 0.204·24-s + 1.45·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.540325299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.540325299\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 17.5T + 125T^{2} \) |
| 11 | \( 1 + 3.08T + 1.33e3T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 126.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 25.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 370.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 461.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 478.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 503.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 681.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 96.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 489.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 271.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 529.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 910.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 305.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−10.33821777560869723513486138236, −9.838042566109044116423786882079, −8.776572339520877250044150411411, −7.34014146228772072466719811718, −6.41058286286950942648420836216, −5.69511202612237312662261677269, −5.10262002890042975946948494123, −3.64249027771704366360354806105, −2.38667019151206458490095094702, −1.15821639174823189992608327213,
1.15821639174823189992608327213, 2.38667019151206458490095094702, 3.64249027771704366360354806105, 5.10262002890042975946948494123, 5.69511202612237312662261677269, 6.41058286286950942648420836216, 7.34014146228772072466719811718, 8.776572339520877250044150411411, 9.838042566109044116423786882079, 10.33821777560869723513486138236
