L(s) = 1 | + 0.635·3-s + 5-s − 4.00·7-s − 2.59·9-s − 4.41·11-s + 3.62·13-s + 0.635·15-s + 3.41·17-s + 1.43·19-s − 2.54·21-s − 6.49·23-s + 25-s − 3.55·27-s − 5.34·29-s − 6.86·31-s − 2.80·33-s − 4.00·35-s + 1.03·37-s + 2.30·39-s + 6.50·41-s + 3.18·43-s − 2.59·45-s + 0.367·47-s + 9.01·49-s + 2.17·51-s + 0.243·53-s − 4.41·55-s + ⋯ |
L(s) = 1 | + 0.366·3-s + 0.447·5-s − 1.51·7-s − 0.865·9-s − 1.33·11-s + 1.00·13-s + 0.164·15-s + 0.828·17-s + 0.329·19-s − 0.554·21-s − 1.35·23-s + 0.200·25-s − 0.684·27-s − 0.993·29-s − 1.23·31-s − 0.488·33-s − 0.676·35-s + 0.170·37-s + 0.369·39-s + 1.01·41-s + 0.486·43-s − 0.387·45-s + 0.0535·47-s + 1.28·49-s + 0.304·51-s + 0.0334·53-s − 0.595·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.255648235\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.255648235\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 0.635T + 3T^{2} \) |
| 7 | \( 1 + 4.00T + 7T^{2} \) |
| 11 | \( 1 + 4.41T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 - 3.41T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 23 | \( 1 + 6.49T + 23T^{2} \) |
| 29 | \( 1 + 5.34T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 6.50T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 - 0.367T + 47T^{2} \) |
| 53 | \( 1 - 0.243T + 53T^{2} \) |
| 59 | \( 1 - 1.72T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 0.802T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 6.85T + 73T^{2} \) |
| 79 | \( 1 + 2.80T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 7.32T + 89T^{2} \) |
| 97 | \( 1 + 8.60T + 97T^{2} \) |
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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
−7.84948207533842218119064496336, −7.23266334053984393417342663188, −6.26553261851880124803792014343, −5.69014789223688481571077942249, −5.45785627434517371298113889524, −3.98822272170683399946482713995, −3.40769101623712800164694410573, −2.76745910415886944298628685540, −1.99450964298813226011767870745, −0.51129606767622360308056533421,
0.51129606767622360308056533421, 1.99450964298813226011767870745, 2.76745910415886944298628685540, 3.40769101623712800164694410573, 3.98822272170683399946482713995, 5.45785627434517371298113889524, 5.69014789223688481571077942249, 6.26553261851880124803792014343, 7.23266334053984393417342663188, 7.84948207533842218119064496336