| L(s) = 1 | − 5.29·5-s − 7.48i·7-s − 5.65i·11-s − 4·13-s + 21.1·17-s − 29.9i·19-s − 22.6i·23-s + 3.00·25-s + 5.29·29-s + 22.4i·31-s + 39.5i·35-s − 28·37-s + 63.4·41-s − 29.9i·43-s + 67.8i·47-s + ⋯ |
| L(s) = 1 | − 1.05·5-s − 1.06i·7-s − 0.514i·11-s − 0.307·13-s + 1.24·17-s − 1.57i·19-s − 0.983i·23-s + 0.120·25-s + 0.182·29-s + 0.724i·31-s + 1.13i·35-s − 0.756·37-s + 1.54·41-s − 0.696i·43-s + 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7675133554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7675133554\) |
| \(L(2)\) |
|
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 | \( 1 + 5.29T + 25T^{2} \) |
| 7 | \( 1 + 7.48iT - 49T^{2} \) |
| 11 | \( 1 + 5.65iT - 121T^{2} \) |
| 13 | \( 1 + 4T + 169T^{2} \) |
| 17 | \( 1 - 21.1T + 289T^{2} \) |
| 19 | \( 1 + 29.9iT - 361T^{2} \) |
| 23 | \( 1 + 22.6iT - 529T^{2} \) |
| 29 | \( 1 - 5.29T + 841T^{2} \) |
| 31 | \( 1 - 22.4iT - 961T^{2} \) |
| 37 | \( 1 + 28T + 1.36e3T^{2} \) |
| 41 | \( 1 - 63.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 29.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76T + 3.72e3T^{2} \) |
| 67 | \( 1 - 59.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 26T + 5.32e3T^{2} \) |
| 79 | \( 1 + 127. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 42.3T + 7.92e3T^{2} \) |
| 97 | \( 1 - 18T + 9.40e3T^{2} \) |
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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
−8.334558300508504523010370584211, −7.55886577259424093398181607189, −7.14961684644089754871899837158, −6.21677256639408718610722215269, −5.06822274678705904002383288159, −4.33943079033002599979898731955, −3.56486009815332256011269847509, −2.72274782627156850661667108940, −1.03657755234955879368478042388, −0.22558777061047336091800918224,
1.38689434686113213569854057134, 2.55798766956239175810603131567, 3.57795477640566694159895759024, 4.25045981991544227666341731161, 5.47461825129319059647224827109, 5.82908937713698727993330295431, 7.11823979730932716233649446437, 7.77647170958900311838092198189, 8.248321163854890725486490421877, 9.222425280656852042824959440135
