| L(s) = 1 | − 1.23e4·3-s − 7.38e5·5-s − 1.17e7·7-s + 2.30e7·9-s + 1.10e9·11-s − 2.57e9·13-s + 9.10e9·15-s + 4.96e10·17-s + 7.58e10·19-s + 1.45e11·21-s + 4.80e10·23-s − 2.17e11·25-s + 1.30e12·27-s − 4.46e12·29-s − 2.89e12·31-s − 1.35e13·33-s + 8.68e12·35-s + 1.98e13·37-s + 3.17e13·39-s − 5.72e13·41-s + 1.17e14·43-s − 1.69e13·45-s + 1.99e14·47-s − 9.42e13·49-s − 6.12e14·51-s + 3.19e14·53-s − 8.13e14·55-s + ⋯ |
| L(s) = 1 | − 1.08·3-s − 0.845·5-s − 0.771·7-s + 0.178·9-s + 1.55·11-s − 0.875·13-s + 0.917·15-s + 1.72·17-s + 1.02·19-s + 0.837·21-s + 0.127·23-s − 0.285·25-s + 0.892·27-s − 1.65·29-s − 0.610·31-s − 1.68·33-s + 0.652·35-s + 0.926·37-s + 0.950·39-s − 1.11·41-s + 1.53·43-s − 0.150·45-s + 1.22·47-s − 0.405·49-s − 1.87·51-s + 0.704·53-s − 1.31·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.8474698493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8474698493\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 1.23e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 7.38e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.17e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.10e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 2.57e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 4.96e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 7.58e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 4.80e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 4.46e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.89e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.98e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 5.72e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.17e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.99e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 3.19e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 3.92e14T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.54e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 2.36e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.35e14T + 2.96e31T^{2} \) |
| 73 | \( 1 - 7.70e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 8.20e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.50e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.99e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 5.80e16T + 5.95e33T^{2} \) |
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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
−17.11134775821291271650384649233, −16.32051997669582610363263362513, −14.56586339369707687320254082529, −12.30524366856057884766205686639, −11.51859337152374843108861076324, −9.613962185338895974396903101037, −7.28213934815970097542690976075, −5.65579569706738419376822699308, −3.66348262963800156090830317223, −0.72011757457027672599701278466,
0.72011757457027672599701278466, 3.66348262963800156090830317223, 5.65579569706738419376822699308, 7.28213934815970097542690976075, 9.613962185338895974396903101037, 11.51859337152374843108861076324, 12.30524366856057884766205686639, 14.56586339369707687320254082529, 16.32051997669582610363263362513, 17.11134775821291271650384649233
