| L(s) = 1 | − 256·2-s + 8.00e3·3-s + 6.55e4·4-s + 3.90e5·5-s − 2.04e6·6-s + 2.54e7·7-s − 1.67e7·8-s − 6.50e7·9-s − 1.00e8·10-s − 2.81e8·11-s + 5.24e8·12-s − 1.52e9·13-s − 6.52e9·14-s + 3.12e9·15-s + 4.29e9·16-s + 5.46e10·17-s + 1.66e10·18-s + 6.88e8·19-s + 2.56e10·20-s + 2.03e11·21-s + 7.19e10·22-s + 3.91e11·23-s − 1.34e11·24-s + 1.52e11·25-s + 3.90e11·26-s − 1.55e12·27-s + 1.66e12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.704·3-s + 0.5·4-s + 0.447·5-s − 0.498·6-s + 1.66·7-s − 0.353·8-s − 0.503·9-s − 0.316·10-s − 0.395·11-s + 0.352·12-s − 0.518·13-s − 1.18·14-s + 0.314·15-s + 0.250·16-s + 1.90·17-s + 0.356·18-s + 0.00930·19-s + 0.223·20-s + 1.17·21-s + 0.279·22-s + 1.04·23-s − 0.249·24-s + 0.200·25-s + 0.366·26-s − 1.05·27-s + 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.156261583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.156261583\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 256T \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 3 | \( 1 - 8.00e3T + 1.29e8T^{2} \) |
| 7 | \( 1 - 2.54e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 2.81e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.52e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 5.46e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 6.88e8T + 5.48e21T^{2} \) |
| 23 | \( 1 - 3.91e11T + 1.41e23T^{2} \) |
| 29 | \( 1 - 5.12e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 7.31e10T + 2.25e25T^{2} \) |
| 37 | \( 1 + 6.81e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.76e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 7.57e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 4.60e13T + 2.66e28T^{2} \) |
| 53 | \( 1 - 6.58e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.98e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 8.50e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 6.12e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 5.41e14T + 2.96e31T^{2} \) |
| 73 | \( 1 + 7.16e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 5.45e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.64e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 6.81e14T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.20e17T + 5.95e33T^{2} \) |
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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
−16.91331647796161257372432449623, −14.92114101550407849727475476248, −14.06847169365456172570887139250, −11.86821136002649854971542806552, −10.32654301785782703697713516185, −8.670268548833463173601824657424, −7.66158257804266450132459066986, −5.26730866841669086707818893707, −2.72388181303130273758290534286, −1.25154400183761327838886927625,
1.25154400183761327838886927625, 2.72388181303130273758290534286, 5.26730866841669086707818893707, 7.66158257804266450132459066986, 8.670268548833463173601824657424, 10.32654301785782703697713516185, 11.86821136002649854971542806552, 14.06847169365456172570887139250, 14.92114101550407849727475476248, 16.91331647796161257372432449623
