| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 13-s + 15-s + 17-s + 4·21-s + 25-s − 27-s + 2·29-s + 4·31-s + 4·35-s + 2·37-s + 39-s − 6·41-s − 4·43-s − 45-s − 12·47-s + 9·49-s − 51-s − 6·53-s + 8·59-s − 14·61-s − 4·63-s + 65-s − 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 0.242·17-s + 0.872·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 9/7·49-s − 0.140·51-s − 0.824·53-s + 1.04·59-s − 1.79·61-s − 0.503·63-s + 0.124·65-s − 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
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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
−13.41372049663209, −12.84351661960965, −12.63599256673146, −12.00769976516770, −11.66591799214463, −11.32433381165325, −10.42571615772718, −10.28511222614503, −9.853664366236003, −9.291426151148722, −8.866484252317326, −8.216724935106040, −7.720501809284792, −7.196208470453297, −6.663412709607966, −6.297951120833460, −5.956997564687868, −5.145953114953811, −4.775239258963852, −4.172589504980061, −3.520582624976604, −3.099499193595944, −2.645921084164993, −1.688034439532584, −1.051224708813212, 0, 0,
1.051224708813212, 1.688034439532584, 2.645921084164993, 3.099499193595944, 3.520582624976604, 4.172589504980061, 4.775239258963852, 5.145953114953811, 5.956997564687868, 6.297951120833460, 6.663412709607966, 7.196208470453297, 7.720501809284792, 8.216724935106040, 8.866484252317326, 9.291426151148722, 9.853664366236003, 10.28511222614503, 10.42571615772718, 11.32433381165325, 11.66591799214463, 12.00769976516770, 12.63599256673146, 12.84351661960965, 13.41372049663209
