L(s) = 1 | + 1.10·3-s − 5-s + 4.43·7-s − 1.76·9-s + 3.25·11-s − 1.74·13-s − 1.10·15-s − 0.929·17-s − 1.73·19-s + 4.92·21-s − 5.06·23-s + 25-s − 5.29·27-s + 7.58·29-s + 1.53·31-s + 3.60·33-s − 4.43·35-s − 11.9·37-s − 1.93·39-s + 9.34·41-s + 6.11·43-s + 1.76·45-s + 2.98·47-s + 12.7·49-s − 1.03·51-s − 5.54·53-s − 3.25·55-s + ⋯ |
L(s) = 1 | + 0.640·3-s − 0.447·5-s + 1.67·7-s − 0.589·9-s + 0.980·11-s − 0.483·13-s − 0.286·15-s − 0.225·17-s − 0.397·19-s + 1.07·21-s − 1.05·23-s + 0.200·25-s − 1.01·27-s + 1.40·29-s + 0.276·31-s + 0.628·33-s − 0.750·35-s − 1.97·37-s − 0.309·39-s + 1.45·41-s + 0.932·43-s + 0.263·45-s + 0.435·47-s + 1.81·49-s − 0.144·51-s − 0.760·53-s − 0.438·55-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
\(L(1)\) |
\(\approx\) |
\(2.849461895\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.849461895\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.10T + 3T^{2} \) |
| 7 | \( 1 - 4.43T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 + 0.929T + 17T^{2} \) |
| 19 | \( 1 + 1.73T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 - 7.58T + 29T^{2} \) |
| 31 | \( 1 - 1.53T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 9.34T + 41T^{2} \) |
| 43 | \( 1 - 6.11T + 43T^{2} \) |
| 47 | \( 1 - 2.98T + 47T^{2} \) |
| 53 | \( 1 + 5.54T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 9.97T + 79T^{2} \) |
| 83 | \( 1 + 3.52T + 83T^{2} \) |
| 89 | \( 1 + 7.10T + 89T^{2} \) |
| 97 | \( 1 + 1.90T + 97T^{2} \) |
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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
−7.968810031885590933696880612399, −7.33269301949278918613233817732, −6.55667644422317137263448456187, −5.66313404646827038418823399353, −4.91828410114990079806357467206, −4.21892430279677423215298347293, −3.65225734126377219780499619893, −2.49236108004283199799429053636, −1.94731749811500106634198338069, −0.821089911916580736654264000028,
0.821089911916580736654264000028, 1.94731749811500106634198338069, 2.49236108004283199799429053636, 3.65225734126377219780499619893, 4.21892430279677423215298347293, 4.91828410114990079806357467206, 5.66313404646827038418823399353, 6.55667644422317137263448456187, 7.33269301949278918613233817732, 7.968810031885590933696880612399