| L(s) = 1 | − 9.07·3-s − 6.17·5-s + 34.5·7-s + 55.3·9-s + 37.1·11-s − 39.9·13-s + 56.0·15-s + 36.3·17-s + 32.0·19-s − 313.·21-s − 202.·23-s − 86.8·25-s − 257.·27-s − 29·29-s − 54.2·31-s − 336.·33-s − 213.·35-s + 221.·37-s + 362.·39-s + 52.3·41-s − 209.·43-s − 341.·45-s − 456.·47-s + 847.·49-s − 330.·51-s + 619.·53-s − 229.·55-s + ⋯ |
| L(s) = 1 | − 1.74·3-s − 0.552·5-s + 1.86·7-s + 2.04·9-s + 1.01·11-s − 0.852·13-s + 0.965·15-s + 0.518·17-s + 0.387·19-s − 3.25·21-s − 1.83·23-s − 0.694·25-s − 1.83·27-s − 0.185·29-s − 0.314·31-s − 1.77·33-s − 1.02·35-s + 0.985·37-s + 1.48·39-s + 0.199·41-s − 0.743·43-s − 1.13·45-s − 1.41·47-s + 2.47·49-s − 0.906·51-s + 1.60·53-s − 0.562·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 + 9.07T + 27T^{2} \) |
| 5 | \( 1 + 6.17T + 125T^{2} \) |
| 7 | \( 1 - 34.5T + 343T^{2} \) |
| 11 | \( 1 - 37.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 39.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 202.T + 1.21e4T^{2} \) |
| 31 | \( 1 + 54.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 52.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 456.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 619.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 878.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 211.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 109.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.20e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 286.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 473.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.03e3T + 9.12e5T^{2} \) |
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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
−8.156774943680153008678750605182, −7.69469906199012538756661666341, −6.86768113384365264226636555877, −5.89163029454459527808970552514, −5.27959865578861643792132960197, −4.48849755835635007050633100605, −3.94678668271016741209032066036, −1.94170474932744337250049912194, −1.13647946646198047682354371983, 0,
1.13647946646198047682354371983, 1.94170474932744337250049912194, 3.94678668271016741209032066036, 4.48849755835635007050633100605, 5.27959865578861643792132960197, 5.89163029454459527808970552514, 6.86768113384365264226636555877, 7.69469906199012538756661666341, 8.156774943680153008678750605182
