| L(s) = 1 | + 1.41·7-s − 0.191i·11-s + 2.63·13-s − 6.20·17-s − 1.52·19-s − 5.25i·23-s − 0.270·29-s + 6.20i·31-s − 7.61·37-s − 9.22i·41-s + 12.7i·43-s − 3.79i·47-s − 5·49-s + 8.77i·53-s + 10.4i·59-s + ⋯ |
| L(s) = 1 | + 0.534·7-s − 0.0577i·11-s + 0.731·13-s − 1.50·17-s − 0.349·19-s − 1.09i·23-s − 0.0502·29-s + 1.11i·31-s − 1.25·37-s − 1.44i·41-s + 1.94i·43-s − 0.553i·47-s − 0.714·49-s + 1.20i·53-s + 1.36i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6393800527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6393800527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 0.191iT - 11T^{2} \) |
| 13 | \( 1 - 2.63T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 1.52T + 19T^{2} \) |
| 23 | \( 1 + 5.25iT - 23T^{2} \) |
| 29 | \( 1 + 0.270T + 29T^{2} \) |
| 31 | \( 1 - 6.20iT - 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 9.22iT - 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 3.79iT - 47T^{2} \) |
| 53 | \( 1 - 8.77iT - 53T^{2} \) |
| 59 | \( 1 - 10.4iT - 59T^{2} \) |
| 61 | \( 1 - 0.382iT - 61T^{2} \) |
| 67 | \( 1 - 1.72iT - 67T^{2} \) |
| 71 | \( 1 + 9.72T + 71T^{2} \) |
| 73 | \( 1 + 5.45iT - 73T^{2} \) |
| 79 | \( 1 - 14.3iT - 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 3.56iT - 89T^{2} \) |
| 97 | \( 1 + 7.31iT - 97T^{2} \) |
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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
−8.456388660800203935784362277514, −7.45808833845418641454605148333, −6.72999810329530170746961678443, −6.23372480346535932667058116027, −5.32152881778421451122084005357, −4.57914431349983384602058995357, −4.01637102657994166401686197539, −2.99478961467025484170452512681, −2.13247036356692919975088657282, −1.23197346549825087467927314208,
0.14903675268625692823972079921, 1.56099694597211711779059440032, 2.19396004576758360589734021633, 3.34495622810507375512280206711, 4.04981727365287642220594855597, 4.81181148769869084437126324125, 5.50078639935567749618273260770, 6.36352574175949316793197647128, 6.86112888821149682033728095801, 7.76672805793416283711870559700
