| L(s) = 1 | − 4.11i·5-s + 7-s − 4.11i·11-s − 5.46i·13-s − 1.10·17-s − 3.46i·19-s + 7.12·23-s − 11.9·25-s − 6.02i·29-s − 2·31-s − 4.11i·35-s + 11.4i·37-s + 1.10·41-s − 8.92i·43-s + 6.02·47-s + ⋯ |
| L(s) = 1 | − 1.84i·5-s + 0.377·7-s − 1.24i·11-s − 1.51i·13-s − 0.267·17-s − 0.794i·19-s + 1.48·23-s − 2.38·25-s − 1.11i·29-s − 0.359·31-s − 0.695i·35-s + 1.88i·37-s + 0.172·41-s − 1.36i·43-s + 0.878·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.799493328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.799493328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 4.11iT - 5T^{2} \) |
| 11 | \( 1 + 4.11iT - 11T^{2} \) |
| 13 | \( 1 + 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 1.10T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 + 6.02iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 11.4iT - 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 + 8.92iT - 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 6.02iT - 59T^{2} \) |
| 61 | \( 1 - 5.46iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 9.33T + 71T^{2} \) |
| 73 | \( 1 + 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 14.2iT - 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−8.246450209431590038527200970699, −7.68785078545402087659822660245, −6.54770627535415765371769273720, −5.49002278647898785927006888210, −5.28619427575377627610584604227, −4.46651374779984069457644115210, −3.51328522323962144730137259199, −2.50328804854216767026337706020, −1.07458099608866738857233135217, −0.58833532543950895253034684684,
1.72844199022732198797612170248, 2.34809482412785503036125066357, 3.35417849632582853447601526736, 4.13734287582915769299400835771, 4.96048774894047972703642058396, 6.05734400497833176794912989937, 6.74974708824799660729716721194, 7.24833175174406642668189095704, 7.67985028841608421336848700003, 8.969499105379810762091505720608
