| L(s) = 1 | − 4·11-s − 13-s + 17-s − 8·19-s − 8·23-s − 6·29-s + 6·37-s − 6·41-s − 4·43-s + 4·47-s − 7·49-s − 10·53-s + 6·61-s + 2·73-s + 8·79-s − 2·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s − 0.277·13-s + 0.242·17-s − 1.83·19-s − 1.66·23-s − 1.11·29-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s − 49-s − 1.37·53-s + 0.768·61-s + 0.234·73-s + 0.900·79-s − 0.211·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.80212315470223, −12.33093602824672, −11.70916556310988, −11.30409048268847, −10.82168255992507, −10.41108863402866, −9.988271730944089, −9.633224686635836, −9.052193566089080, −8.437643352574007, −8.059525088780484, −7.820054839555887, −7.248224853864765, −6.561851708723507, −6.247773469076428, −5.758224569394487, −5.141970142268044, −4.832241058802119, −4.043354450144174, −3.884405489119845, −3.040763590760380, −2.561040712753905, −1.948053648737684, −1.666666144473750, −0.4883867220292038, 0,
0.4883867220292038, 1.666666144473750, 1.948053648737684, 2.561040712753905, 3.040763590760380, 3.884405489119845, 4.043354450144174, 4.832241058802119, 5.141970142268044, 5.758224569394487, 6.247773469076428, 6.561851708723507, 7.248224853864765, 7.820054839555887, 8.059525088780484, 8.437643352574007, 9.052193566089080, 9.633224686635836, 9.988271730944089, 10.41108863402866, 10.82168255992507, 11.30409048268847, 11.70916556310988, 12.33093602824672, 12.80212315470223
