| L(s) = 1 | + 3.69·5-s − i·7-s + 2.05i·11-s − 4.65i·13-s − 5.73i·17-s − 3.27·19-s − 4.45·23-s + 8.66·25-s − 10.2·29-s − 1.26i·31-s − 3.69i·35-s − 3.25i·37-s − 8.72i·41-s + 2.56·43-s + 2.03·47-s + ⋯ |
| L(s) = 1 | + 1.65·5-s − 0.377i·7-s + 0.620i·11-s − 1.28i·13-s − 1.39i·17-s − 0.750·19-s − 0.928·23-s + 1.73·25-s − 1.90·29-s − 0.226i·31-s − 0.624i·35-s − 0.535i·37-s − 1.36i·41-s + 0.391·43-s + 0.296·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736408042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736408042\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 - 2.05iT - 11T^{2} \) |
| 13 | \( 1 + 4.65iT - 13T^{2} \) |
| 17 | \( 1 + 5.73iT - 17T^{2} \) |
| 19 | \( 1 + 3.27T + 19T^{2} \) |
| 23 | \( 1 + 4.45T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.26iT - 31T^{2} \) |
| 37 | \( 1 + 3.25iT - 37T^{2} \) |
| 41 | \( 1 + 8.72iT - 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 2.03T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 0.851iT - 59T^{2} \) |
| 61 | \( 1 - 2.31iT - 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + 6.97T + 73T^{2} \) |
| 79 | \( 1 + 11.6iT - 79T^{2} \) |
| 83 | \( 1 - 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 2.35iT - 89T^{2} \) |
| 97 | \( 1 - 4.02T + 97T^{2} \) |
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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
−7.55831475513020175479268120893, −7.29720600003949948347097645199, −6.20920378226991828367751603715, −5.74523443283049524390609126772, −5.12856150138385868370571919612, −4.27638833531119017735434879275, −3.20161974334436224383202072858, −2.32118944200986004464465115032, −1.70146802340549629309170916432, −0.37184361073096401417524443320,
1.66874583754443467503229268904, 1.84347224376130460011251938659, 2.93108526535656183772164849187, 3.96829869305781368091254157386, 4.74775683096026744323799738730, 5.80529638527368780024974000637, 6.05790982591678072211871740110, 6.58111441943891266733749751350, 7.62850963065032893717969677709, 8.523517470153322018987860091770
