| L(s) = 1 | + 0.438·2-s − 3.68·3-s − 7.80·4-s + 17.8·5-s − 1.61·6-s − 5.43·7-s − 6.93·8-s − 13.4·9-s + 7.80·10-s + 22.4·11-s + 28.7·12-s − 2.38·14-s − 65.6·15-s + 59.4·16-s + 67.9·17-s − 5.88·18-s + 80.8·19-s − 139.·20-s + 20.0·21-s + 9.83·22-s + 140.·23-s + 25.5·24-s + 192.·25-s + 148.·27-s + 42.4·28-s − 106.·29-s − 28.7·30-s + ⋯ |
| L(s) = 1 | + 0.155·2-s − 0.709·3-s − 0.975·4-s + 1.59·5-s − 0.109·6-s − 0.293·7-s − 0.306·8-s − 0.497·9-s + 0.246·10-s + 0.614·11-s + 0.692·12-s − 0.0455·14-s − 1.12·15-s + 0.928·16-s + 0.969·17-s − 0.0770·18-s + 0.975·19-s − 1.55·20-s + 0.208·21-s + 0.0952·22-s + 1.27·23-s + 0.217·24-s + 1.53·25-s + 1.06·27-s + 0.286·28-s − 0.683·29-s − 0.175·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.472356859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472356859\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 0.438T + 8T^{2} \) |
| 3 | \( 1 + 3.68T + 27T^{2} \) |
| 5 | \( 1 - 17.8T + 125T^{2} \) |
| 7 | \( 1 + 5.43T + 343T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 17 | \( 1 - 67.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 4.29T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 27.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 67.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 663.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 152.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 117.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 202.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 336.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 718.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 759.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.46757631361044488655896404301, −11.38626481682527784589222778121, −10.01043131015947853918781270110, −9.555105362679127897517757011849, −8.473443844211579922125131180053, −6.64477765530419764883446938504, −5.66154495302004080509277038735, −5.02837983571621671716504499245, −3.14963945249833217200481794967, −1.05122902065604619535772741546,
1.05122902065604619535772741546, 3.14963945249833217200481794967, 5.02837983571621671716504499245, 5.66154495302004080509277038735, 6.64477765530419764883446938504, 8.473443844211579922125131180053, 9.555105362679127897517757011849, 10.01043131015947853918781270110, 11.38626481682527784589222778121, 12.46757631361044488655896404301
