L(s) = 1 | + (−1 − 3.73i)5-s + (−0.633 + 0.366i)7-s + (2.86 + 0.767i)11-s + (6.09 − 1.63i)13-s + 2.26·17-s + (0.633 + 0.633i)19-s + (1.09 + 0.633i)23-s + (−8.59 + 4.96i)25-s + (−0.633 + 2.36i)29-s + (3.73 − 6.46i)31-s + (2 + 2i)35-s + (1.26 − 1.26i)37-s + (−2.59 − 1.5i)41-s + (1.23 + 0.330i)43-s + (−4.83 − 8.36i)47-s + ⋯ |
L(s) = 1 | + (−0.447 − 1.66i)5-s + (−0.239 + 0.138i)7-s + (0.864 + 0.231i)11-s + (1.69 − 0.453i)13-s + 0.550·17-s + (0.145 + 0.145i)19-s + (0.228 + 0.132i)23-s + (−1.71 + 0.992i)25-s + (−0.117 + 0.439i)29-s + (0.670 − 1.16i)31-s + (0.338 + 0.338i)35-s + (0.208 − 0.208i)37-s + (−0.405 − 0.234i)41-s + (0.187 + 0.0503i)43-s + (−0.704 − 1.22i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0436 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.694137891\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.694137891\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 + 3.73i)T + (-4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.86 - 0.767i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-6.09 + 1.63i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 - 2.26T + 17T^{2} \) |
| 19 | \( 1 + (-0.633 - 0.633i)T + 19iT^{2} \) |
| 23 | \( 1 + (-1.09 - 0.633i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.633 - 2.36i)T + (-25.1 - 14.5i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 6.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.26 + 1.26i)T - 37iT^{2} \) |
| 41 | \( 1 + (2.59 + 1.5i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.23 - 0.330i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (4.83 + 8.36i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.535 + 0.535i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.33 + 4.96i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.803 + 3i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-5.23 + 1.40i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 + 9.73iT - 73T^{2} \) |
| 79 | \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.366 + 1.36i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 2iT - 89T^{2} \) |
| 97 | \( 1 + (4.13 + 7.16i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.152761941898776517771323275547, −8.298059296746447792189479829794, −7.911204701159158097871661665197, −6.61566987415076685668379069099, −5.80621193273658517338743310043, −5.02388676061866932290372354591, −4.06305278238611565285353246796, −3.43193989464169651925909760574, −1.62724581939789917924057996070, −0.75824261685559736102049353203,
1.31933835114163033168861052078, 2.85272185069481697134385075933, 3.52233089182616022207573523700, 4.21977933441408093319507163962, 5.73878116097684680097118464735, 6.61066232324852475097844279835, 6.80449250273389071687045354783, 7.922496211004158647687211911843, 8.634299490260216176265056361699, 9.620756770168070227199172271429