| L(s) = 1 | + 8.16·3-s + 4.64·5-s + 13.4·7-s + 39.7·9-s + 13.2·11-s − 24.5·13-s + 37.9·15-s − 37.4·19-s + 109.·21-s + 45.9·23-s − 103.·25-s + 103.·27-s − 0.881·29-s − 27.3·31-s + 107.·33-s + 62.1·35-s + 267.·37-s − 200.·39-s + 444.·41-s + 405.·43-s + 184.·45-s + 421.·47-s − 163.·49-s − 264.·53-s + 61.2·55-s − 306.·57-s − 77.7·59-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.415·5-s + 0.723·7-s + 1.47·9-s + 0.361·11-s − 0.523·13-s + 0.652·15-s − 0.452·19-s + 1.13·21-s + 0.416·23-s − 0.827·25-s + 0.740·27-s − 0.00564·29-s − 0.158·31-s + 0.568·33-s + 0.300·35-s + 1.18·37-s − 0.823·39-s + 1.69·41-s + 1.43·43-s + 0.610·45-s + 1.30·47-s − 0.476·49-s − 0.685·53-s + 0.150·55-s − 0.711·57-s − 0.171·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.347565961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.347565961\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 8.16T + 27T^{2} \) |
| 5 | \( 1 - 4.64T + 125T^{2} \) |
| 7 | \( 1 - 13.4T + 343T^{2} \) |
| 11 | \( 1 - 13.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.5T + 2.19e3T^{2} \) |
| 19 | \( 1 + 37.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 45.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.881T + 2.43e4T^{2} \) |
| 31 | \( 1 + 27.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 421.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 264.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 77.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 874.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 46.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 941.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 157.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 911.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 89.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 778.T + 9.12e5T^{2} \) |
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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
−8.665445677492978484114557950017, −7.88209631623055771490631254116, −7.48244282246589758680835288825, −6.43182983789616508869742545408, −5.44539726149060500040350121439, −4.38549191254972192806740369733, −3.76952464456701691169415643786, −2.55662585682133304634354098319, −2.14288173323631169498920753439, −0.992964115967573107092048070202,
0.992964115967573107092048070202, 2.14288173323631169498920753439, 2.55662585682133304634354098319, 3.76952464456701691169415643786, 4.38549191254972192806740369733, 5.44539726149060500040350121439, 6.43182983789616508869742545408, 7.48244282246589758680835288825, 7.88209631623055771490631254116, 8.665445677492978484114557950017
