| L(s) = 1 | + 5-s − 13-s − 4·17-s − 6·19-s − 6·23-s + 25-s − 4·29-s − 8·31-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s − 2·53-s + 2·61-s − 65-s − 4·67-s + 8·71-s + 16·79-s − 4·83-s − 4·85-s − 6·89-s − 6·95-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.277·13-s − 0.970·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 0.742·29-s − 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 0.274·53-s + 0.256·61-s − 0.124·65-s − 0.488·67-s + 0.949·71-s + 1.80·79-s − 0.439·83-s − 0.433·85-s − 0.635·89-s − 0.615·95-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.06111912253740, −12.68933629183560, −12.40896181719607, −11.79128182506265, −11.19639459163950, −10.70291667027289, −10.61202388398431, −9.816767290578042, −9.448636626576803, −8.908863660169583, −8.620064281133137, −7.955027870291320, −7.478088836474440, −6.942989581615446, −6.496796082941444, −5.893317687402635, −5.611385793281580, −4.943624137507778, −4.265665887801836, −4.012665769805623, −3.348433368121441, −2.525866945341528, −2.007595279176981, −1.810855697055542, −0.6648579686446148, 0,
0.6648579686446148, 1.810855697055542, 2.007595279176981, 2.525866945341528, 3.348433368121441, 4.012665769805623, 4.265665887801836, 4.943624137507778, 5.611385793281580, 5.893317687402635, 6.496796082941444, 6.942989581615446, 7.478088836474440, 7.955027870291320, 8.620064281133137, 8.908863660169583, 9.448636626576803, 9.816767290578042, 10.61202388398431, 10.70291667027289, 11.19639459163950, 11.79128182506265, 12.40896181719607, 12.68933629183560, 13.06111912253740
