| L(s) = 1 | + 2i·5-s − 6.92i·7-s − 6.92·11-s − 2i·13-s − 10·17-s − 20.7·19-s + 27.7i·23-s + 21·25-s − 26i·29-s − 6.92i·31-s + 13.8·35-s + 26i·37-s + 58·41-s − 48.4·43-s + 69.2i·47-s + ⋯ |
| L(s) = 1 | + 0.400i·5-s − 0.989i·7-s − 0.629·11-s − 0.153i·13-s − 0.588·17-s − 1.09·19-s + 1.20i·23-s + 0.839·25-s − 0.896i·29-s − 0.223i·31-s + 0.395·35-s + 0.702i·37-s + 1.41·41-s − 1.12·43-s + 1.47i·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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.469264670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.469264670\) |
| \(L(2)\) |
|
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 | \( 1 - 2iT - 25T^{2} \) |
| 7 | \( 1 + 6.92iT - 49T^{2} \) |
| 11 | \( 1 + 6.92T + 121T^{2} \) |
| 13 | \( 1 + 2iT - 169T^{2} \) |
| 17 | \( 1 + 10T + 289T^{2} \) |
| 19 | \( 1 + 20.7T + 361T^{2} \) |
| 23 | \( 1 - 27.7iT - 529T^{2} \) |
| 29 | \( 1 + 26iT - 841T^{2} \) |
| 31 | \( 1 + 6.92iT - 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 58T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 69.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 74iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 90.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 26iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 6.92T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 46T + 5.32e3T^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 48.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 82T + 7.92e3T^{2} \) |
| 97 | \( 1 - 2T + 9.40e3T^{2} \) |
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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
−8.926504236326424102450359655743, −7.977909688338789284176002652868, −7.45200301328249292131828950185, −6.63042287621940803371857315765, −5.91838537362366870714874660656, −4.80278860746713486783798607421, −4.11345153784203473493615048325, −3.13586598963087078092760774355, −2.15957543020744756516764245928, −0.823185506648304492654521347824,
0.45895185726192935375148464568, 2.01791558252308334864128376231, 2.66379287857066629605144509506, 3.91480306672593296944873825757, 4.88075447761143464262489334074, 5.44679344780257172217878757204, 6.44345903248389114371277421086, 7.05219982791205142607729206790, 8.357807089903470264188236802150, 8.565872882739516553692511289684
