| L(s) = 1 | + 4.92i·5-s + 2.92i·7-s + 14.9·11-s + 23.8i·13-s + 19.8·17-s − 30.9·19-s + 8i·23-s + 0.712·25-s + 16.9i·29-s + 38.6i·31-s − 14.4·35-s + 9.71i·37-s − 8.14·41-s + 5.35·43-s − 69.8i·47-s + ⋯ |
| L(s) = 1 | + 0.985i·5-s + 0.418i·7-s + 1.35·11-s + 1.83i·13-s + 1.16·17-s − 1.62·19-s + 0.347i·23-s + 0.0285·25-s + 0.583i·29-s + 1.24i·31-s − 0.412·35-s + 0.262i·37-s − 0.198·41-s + 0.124·43-s − 1.48i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.992540880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.992540880\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.92iT - 25T^{2} \) |
| 7 | \( 1 - 2.92iT - 49T^{2} \) |
| 11 | \( 1 - 14.9T + 121T^{2} \) |
| 13 | \( 1 - 23.8iT - 169T^{2} \) |
| 17 | \( 1 - 19.8T + 289T^{2} \) |
| 19 | \( 1 + 30.9T + 361T^{2} \) |
| 23 | \( 1 - 8iT - 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 38.6iT - 961T^{2} \) |
| 37 | \( 1 - 9.71iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 8.14T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.35T + 1.84e3T^{2} \) |
| 47 | \( 1 + 69.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.928iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 108.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 14iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 16.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 43.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 62.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 50.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 145.T + 9.40e3T^{2} \) |
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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
−8.939443278968734656088384860420, −8.639796339951427307177043489080, −7.32234462895720243383377650302, −6.67289107877763796803422128684, −6.34297505982050313148817433321, −5.17070008928546228747172148835, −4.09702093406016090965031493627, −3.48212090026721551318777920001, −2.29698837811172591115237654092, −1.42834492071673055946970751512,
0.52980784046990390689277323513, 1.23890265910240499675764456226, 2.62788513458911738723256341749, 3.83592096661042773460359351417, 4.37233411740529228368270218376, 5.46931319029437720126099916950, 6.03808616644969536814170447552, 7.02301222473658973431035991844, 8.009266060269973699138290759508, 8.405760058149692612079604181528
