L(s) = 1 | + (0.486 − 1.32i)2-s − 0.812·3-s + (−1.52 − 1.29i)4-s − i·5-s + (−0.395 + 1.07i)6-s + (−2.45 + 1.39i)8-s − 2.34·9-s + (−1.32 − 0.486i)10-s − 4.86i·11-s + (1.23 + 1.04i)12-s + 0.895i·13-s + 0.812i·15-s + (0.658 + 3.94i)16-s + 5.89i·17-s + (−1.13 + 3.10i)18-s − 2.91·19-s + ⋯ |
L(s) = 1 | + (0.344 − 0.938i)2-s − 0.468·3-s + (−0.763 − 0.646i)4-s − 0.447i·5-s + (−0.161 + 0.440i)6-s + (−0.869 + 0.494i)8-s − 0.780·9-s + (−0.419 − 0.153i)10-s − 1.46i·11-s + (0.357 + 0.302i)12-s + 0.248i·13-s + 0.209i·15-s + (0.164 + 0.986i)16-s + 1.42i·17-s + (−0.268 + 0.732i)18-s − 0.669·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0110 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0110 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0708303 + 0.0716161i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0708303 + 0.0716161i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.486 + 1.32i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.812T + 3T^{2} \) |
| 11 | \( 1 + 4.86iT - 11T^{2} \) |
| 13 | \( 1 - 0.895iT - 13T^{2} \) |
| 17 | \( 1 - 5.89iT - 17T^{2} \) |
| 19 | \( 1 + 2.91T + 19T^{2} \) |
| 23 | \( 1 + 1.56iT - 23T^{2} \) |
| 29 | \( 1 + 9.73T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 + 1.82T + 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 3.04iT - 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 0.179T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 6.15iT - 61T^{2} \) |
| 67 | \( 1 + 7.79iT - 67T^{2} \) |
| 71 | \( 1 - 8.38iT - 71T^{2} \) |
| 73 | \( 1 + 8.78iT - 73T^{2} \) |
| 79 | \( 1 - 3.63iT - 79T^{2} \) |
| 83 | \( 1 - 2.03T + 83T^{2} \) |
| 89 | \( 1 + 1.76iT - 89T^{2} \) |
| 97 | \( 1 + 7.83iT - 97T^{2} \) |
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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
−9.458454745378026668509883547113, −8.602147632217234237133894604947, −8.149666267574933546456038521942, −6.22324454970133621593438104233, −5.93455383928205352334314206388, −4.89282266744385152498137212138, −3.88145238182561010607809894552, −2.92774194773503904404176407926, −1.50942821092903267038002189601, −0.04360922744047861354344297945,
2.43135570976499798493576260411, 3.65731781569277731383444570397, 4.80889878654573018611653044913, 5.44281358937629575420176394081, 6.42172407338993400124422838130, 7.17798461985891934502947651254, 7.79348240674655613535863612952, 8.950917969718390866300426412741, 9.603471998370721119221547457816, 10.58471526023154811900696924218