L(s) = 1 | + (0.249 − 0.968i)2-s + (−0.211 − 0.977i)3-s + (−0.875 − 0.483i)4-s + (−0.999 − 0.0387i)6-s + (−0.686 + 0.727i)8-s + (−0.910 + 0.413i)9-s + (−1.35 − 1.04i)11-s + (−0.286 + 0.957i)12-s + (0.533 + 0.845i)16-s + (−0.771 − 0.387i)17-s + (0.173 + 0.984i)18-s + (0.0157 − 0.270i)19-s + (−1.35 + 1.04i)22-s + (0.856 + 0.516i)24-s + (−0.565 + 0.824i)25-s + ⋯ |
L(s) = 1 | + (0.249 − 0.968i)2-s + (−0.211 − 0.977i)3-s + (−0.875 − 0.483i)4-s + (−0.999 − 0.0387i)6-s + (−0.686 + 0.727i)8-s + (−0.910 + 0.413i)9-s + (−1.35 − 1.04i)11-s + (−0.286 + 0.957i)12-s + (0.533 + 0.845i)16-s + (−0.771 − 0.387i)17-s + (0.173 + 0.984i)18-s + (0.0157 − 0.270i)19-s + (−1.35 + 1.04i)22-s + (0.856 + 0.516i)24-s + (−0.565 + 0.824i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4500008564\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4500008564\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.249 + 0.968i)T \) |
| 3 | \( 1 + (0.211 + 0.977i)T \) |
good | 5 | \( 1 + (0.565 - 0.824i)T^{2} \) |
| 7 | \( 1 + (0.740 + 0.672i)T^{2} \) |
| 11 | \( 1 + (1.35 + 1.04i)T + (0.249 + 0.968i)T^{2} \) |
| 13 | \( 1 + (0.627 + 0.778i)T^{2} \) |
| 17 | \( 1 + (0.771 + 0.387i)T + (0.597 + 0.802i)T^{2} \) |
| 19 | \( 1 + (-0.0157 + 0.270i)T + (-0.993 - 0.116i)T^{2} \) |
| 23 | \( 1 + (0.211 + 0.977i)T^{2} \) |
| 29 | \( 1 + (0.981 - 0.192i)T^{2} \) |
| 31 | \( 1 + (0.790 + 0.612i)T^{2} \) |
| 37 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 41 | \( 1 + (1.05 - 1.03i)T + (0.0193 - 0.999i)T^{2} \) |
| 43 | \( 1 + (0.302 + 1.39i)T + (-0.910 + 0.413i)T^{2} \) |
| 47 | \( 1 + (0.790 - 0.612i)T^{2} \) |
| 53 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (0.144 + 1.05i)T + (-0.963 + 0.268i)T^{2} \) |
| 61 | \( 1 + (-0.533 + 0.845i)T^{2} \) |
| 67 | \( 1 + (0.0697 - 0.716i)T + (-0.981 - 0.192i)T^{2} \) |
| 71 | \( 1 + (0.835 + 0.549i)T^{2} \) |
| 73 | \( 1 + (-1.58 - 0.375i)T + (0.893 + 0.448i)T^{2} \) |
| 79 | \( 1 + (-0.856 + 0.516i)T^{2} \) |
| 83 | \( 1 + (1.41 + 1.39i)T + (0.0193 + 0.999i)T^{2} \) |
| 89 | \( 1 + (-0.0333 - 0.111i)T + (-0.835 + 0.549i)T^{2} \) |
| 97 | \( 1 + (-0.301 + 0.572i)T + (-0.565 - 0.824i)T^{2} \) |
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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
−8.685707252229951991492235460842, −8.264876468248813765904916893895, −7.30602792829136690870335742270, −6.30502723618728198951246501463, −5.46052337072295782835451277231, −4.90791949276926320123266541412, −3.47684310788733830477252093445, −2.70605281676288582490735445865, −1.77267873582196003121664144183, −0.29214751300694434754645117651,
2.45530158632221433528707812238, 3.60641893670423531609274134836, 4.53541390780258783254652574261, 5.00245827245404963966193052454, 5.89229595854881131340856571480, 6.64169639578413922236356998439, 7.66511433705980260584645482862, 8.270512941131150808682649733669, 9.089182811806118216660166540523, 9.917404645211120498926845997629