| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 0.329i·7-s − i·8-s − 9-s − 5.03·11-s + i·12-s + 0.482i·13-s + 0.329·14-s + 16-s + 6.78i·17-s − i·18-s − 5.44·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.124i·7-s − 0.353i·8-s − 0.333·9-s − 1.51·11-s + 0.288i·12-s + 0.133i·13-s + 0.0880·14-s + 0.250·16-s + 1.64i·17-s − 0.235i·18-s − 1.24·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.241147615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.241147615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 0.329iT - 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 - 0.482iT - 13T^{2} \) |
| 17 | \( 1 - 6.78iT - 17T^{2} \) |
| 19 | \( 1 + 5.44T + 19T^{2} \) |
| 23 | \( 1 + 6.50iT - 23T^{2} \) |
| 29 | \( 1 - 6.02T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 - 0.780iT - 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 - 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 4.38iT - 47T^{2} \) |
| 53 | \( 1 - 1.68iT - 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 4.18T + 61T^{2} \) |
| 67 | \( 1 - 3.14iT - 67T^{2} \) |
| 71 | \( 1 - 5.71T + 71T^{2} \) |
| 73 | \( 1 + 2.94iT - 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 + 3.45T + 89T^{2} \) |
| 97 | \( 1 + 9.51iT - 97T^{2} \) |
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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
−8.459020769647851393438789558369, −7.79510511217146264103192380239, −7.06326191901712249762574782793, −6.19894106474576628537562225752, −5.86657052753033148470951601405, −4.73430111721733276984851607227, −4.15289140936187666507302914164, −2.86015352004541499645155392459, −2.02839903985366106422149008552, −0.54944710975251981141843118293,
0.71365725160788574132299184096, 2.37380744451097850510914899093, 2.78287901563275142397416939457, 3.83049798451025670233066220513, 4.73081763597418649474603702143, 5.25536146123030467180429045140, 6.04121734738222670634749326752, 7.22788570743336130999701952200, 7.901551803949711046177497685157, 8.654230044501821277976147148968
