| L(s) = 1 | + (0.593 − 0.804i)2-s + (1.65 − 0.312i)3-s + (−0.294 − 0.955i)4-s + (2.14 + 0.646i)5-s + (0.729 − 1.51i)6-s + (−1.08 + 2.41i)7-s + (−0.943 − 0.330i)8-s + (−0.159 + 0.0625i)9-s + (1.79 − 1.33i)10-s + (1.58 − 4.03i)11-s + (−0.785 − 1.48i)12-s + (2.32 + 0.262i)13-s + (1.29 + 2.30i)14-s + (3.73 + 0.399i)15-s + (−0.826 + 0.563i)16-s + (0.779 + 0.0291i)17-s + ⋯ |
| L(s) = 1 | + (0.419 − 0.568i)2-s + (0.954 − 0.180i)3-s + (−0.147 − 0.477i)4-s + (0.957 + 0.289i)5-s + (0.297 − 0.618i)6-s + (−0.411 + 0.911i)7-s + (−0.333 − 0.116i)8-s + (−0.0531 + 0.0208i)9-s + (0.566 − 0.423i)10-s + (0.477 − 1.21i)11-s + (−0.226 − 0.429i)12-s + (0.645 + 0.0727i)13-s + (0.345 + 0.616i)14-s + (0.965 + 0.103i)15-s + (−0.206 + 0.140i)16-s + (0.189 + 0.00707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.37703 - 0.979875i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.37703 - 0.979875i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.593 + 0.804i)T \) |
| 5 | \( 1 + (-2.14 - 0.646i)T \) |
| 7 | \( 1 + (1.08 - 2.41i)T \) |
| good | 3 | \( 1 + (-1.65 + 0.312i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-1.58 + 4.03i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-2.32 - 0.262i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (-0.779 - 0.0291i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-3.47 + 6.01i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.123 - 3.30i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (8.83 - 2.01i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.15 - 1.24i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.10 - 2.17i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-0.857 - 1.78i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.471 + 1.34i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (0.730 + 0.539i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (8.97 + 4.74i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.327 - 4.36i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.74 - 8.88i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-13.0 - 3.49i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.703 - 3.08i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (1.23 - 0.914i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (9.34 + 5.39i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.463 + 4.11i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (4.82 + 12.2i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (5.07 + 5.07i)T + 97iT^{2} \) |
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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
−11.11608753881857246315135772358, −9.757444095366821270937518520173, −9.070832124411563819082100170554, −8.612685459463226393146682000761, −7.09773298408052691372104802577, −5.93040419046215921538804942917, −5.36846349442129707333998923724, −3.42765633985486806594347670921, −2.91682246820980112790385858475, −1.69753323039962791657350157925,
1.83514243478313753086594428713, 3.37725684713983210078832709798, 4.18196660982019262865091658466, 5.50910080098522342580015803016, 6.44239069051901437107405155037, 7.45218907037373309292897957409, 8.294098460513505165197622330436, 9.519358822146862690452478553445, 9.659636426975071342769545750972, 10.93028519593262135689080037420
