| L(s) = 1 | + i·3-s − 1.32i·7-s − 9-s − 3.84·11-s + 4.36i·13-s + 2.75i·17-s + 5.92·19-s + 1.32·21-s − i·23-s − i·27-s − 0.761·29-s − 8.72·31-s − 3.84i·33-s + 3.24i·37-s − 4.36·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.499i·7-s − 0.333·9-s − 1.15·11-s + 1.21i·13-s + 0.669i·17-s + 1.35·19-s + 0.288·21-s − 0.208i·23-s − 0.192i·27-s − 0.141·29-s − 1.56·31-s − 0.669i·33-s + 0.532i·37-s − 0.698·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2520413970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2520413970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.32iT - 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 - 4.36iT - 13T^{2} \) |
| 17 | \( 1 - 2.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.92T + 19T^{2} \) |
| 29 | \( 1 + 0.761T + 29T^{2} \) |
| 31 | \( 1 + 8.72T + 31T^{2} \) |
| 37 | \( 1 - 3.24iT - 37T^{2} \) |
| 41 | \( 1 - 4.16T + 41T^{2} \) |
| 43 | \( 1 - 9.72iT - 43T^{2} \) |
| 47 | \( 1 + 2.92iT - 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 - 3.08T + 59T^{2} \) |
| 61 | \( 1 + 6.08T + 61T^{2} \) |
| 67 | \( 1 - 2.19iT - 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 + 9.20iT - 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 3.03iT - 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 5.43iT - 97T^{2} \) |
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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
−8.315086130681236306368257442763, −7.65233690349355420947268268754, −7.07152621552942004650388206368, −6.20200412867593025061850855868, −5.41821247120966990800559829897, −4.81640714122696303015687864904, −4.02401190140712132977252826677, −3.35667907431116049227856361443, −2.41991733068359404362825558133, −1.39022045198088591205284729123,
0.06414555308049292543976963153, 1.14851076615354850937021519043, 2.38461789267553863276502191962, 2.88894860379068169646096111531, 3.74343765044263181822247861239, 5.01629255335716809639138138008, 5.54893812981739745102409757827, 5.87798814904906325767468609303, 7.25167247851772216723323198530, 7.41047962833030892897465496242
