| L(s) = 1 | − 2.92·3-s + 14.3·5-s − 16.8·7-s − 18.4·9-s − 16.5·11-s + 55.3·13-s − 41.9·15-s + 6.28·17-s + 119.·19-s + 49.3·21-s − 22.5·23-s + 80.3·25-s + 132.·27-s − 29·29-s − 228.·31-s + 48.4·33-s − 241.·35-s + 257.·37-s − 162.·39-s + 382.·41-s + 170.·43-s − 264.·45-s + 172.·47-s − 58.0·49-s − 18.3·51-s − 69.2·53-s − 237.·55-s + ⋯ |
| L(s) = 1 | − 0.563·3-s + 1.28·5-s − 0.911·7-s − 0.682·9-s − 0.453·11-s + 1.18·13-s − 0.721·15-s + 0.0896·17-s + 1.43·19-s + 0.513·21-s − 0.204·23-s + 0.643·25-s + 0.947·27-s − 0.185·29-s − 1.32·31-s + 0.255·33-s − 1.16·35-s + 1.14·37-s − 0.665·39-s + 1.45·41-s + 0.603·43-s − 0.875·45-s + 0.536·47-s − 0.169·49-s − 0.0504·51-s − 0.179·53-s − 0.581·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.708029526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708029526\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 2.92T + 27T^{2} \) |
| 5 | \( 1 - 14.3T + 125T^{2} \) |
| 7 | \( 1 + 16.8T + 343T^{2} \) |
| 11 | \( 1 + 16.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 55.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 6.28T + 4.91e3T^{2} \) |
| 19 | \( 1 - 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 22.5T + 1.21e4T^{2} \) |
| 31 | \( 1 + 228.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 257.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 382.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 170.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 172.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 69.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 43.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 684.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 528.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 488.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 80.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 741.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 957.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−10.68768755573994308039951728019, −9.631055540499419699434242906142, −9.177606371934210431711176816796, −7.88084703595962375522281441677, −6.58516092697133747649306096488, −5.83998349493891296861018088557, −5.35310759435832215862349282416, −3.58985184387095464660757613159, −2.44077180323286521361124900261, −0.854832125783655252599219029392,
0.854832125783655252599219029392, 2.44077180323286521361124900261, 3.58985184387095464660757613159, 5.35310759435832215862349282416, 5.83998349493891296861018088557, 6.58516092697133747649306096488, 7.88084703595962375522281441677, 9.177606371934210431711176816796, 9.631055540499419699434242906142, 10.68768755573994308039951728019
