L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s + 13-s − 14-s − 15-s + 16-s + 3·17-s − 18-s − 4·19-s + 20-s − 21-s − 2·22-s + 2·23-s + 24-s − 4·25-s − 26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.218·21-s − 0.426·22-s + 0.417·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71058 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 911 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.33611785151139, −14.04083089764079, −13.21210019871068, −12.91442465683628, −12.09374186045830, −11.84110808697540, −11.42135503134658, −10.66292739412979, −10.35482515277015, −9.969318169695628, −9.280227329180720, −8.778940641453389, −8.362824548642802, −7.676757507899825, −7.203822378730404, −6.538006217282840, −6.093297503536478, −5.707059134082249, −4.917883916876799, −4.353947643276872, −3.736988141706291, −2.852517472488809, −2.264127248121506, −1.381646598313490, −1.052811642502958, 0,
1.052811642502958, 1.381646598313490, 2.264127248121506, 2.852517472488809, 3.736988141706291, 4.353947643276872, 4.917883916876799, 5.707059134082249, 6.093297503536478, 6.538006217282840, 7.203822378730404, 7.676757507899825, 8.362824548642802, 8.778940641453389, 9.280227329180720, 9.969318169695628, 10.35482515277015, 10.66292739412979, 11.42135503134658, 11.84110808697540, 12.09374186045830, 12.91442465683628, 13.21210019871068, 14.04083089764079, 14.33611785151139