L(s) = 1 | + 3-s − 1.35·5-s − 0.0125·7-s + 9-s − 2.88·11-s + 2.96·13-s − 1.35·15-s − 4.40·17-s − 7.61·19-s − 0.0125·21-s + 4.16·23-s − 3.15·25-s + 27-s + 10.4·29-s + 3.70·31-s − 2.88·33-s + 0.0170·35-s − 2.43·37-s + 2.96·39-s − 3.30·41-s + 12.1·43-s − 1.35·45-s + 3.83·47-s − 6.99·49-s − 4.40·51-s + 0.362·53-s + 3.92·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.607·5-s − 0.00473·7-s + 0.333·9-s − 0.869·11-s + 0.823·13-s − 0.350·15-s − 1.06·17-s − 1.74·19-s − 0.00273·21-s + 0.868·23-s − 0.630·25-s + 0.192·27-s + 1.93·29-s + 0.665·31-s − 0.502·33-s + 0.00288·35-s − 0.399·37-s + 0.475·39-s − 0.515·41-s + 1.85·43-s − 0.202·45-s + 0.558·47-s − 0.999·49-s − 0.616·51-s + 0.0498·53-s + 0.528·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.810274855\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.810274855\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
good | 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 + 0.0125T + 7T^{2} \) |
| 11 | \( 1 + 2.88T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 4.40T + 17T^{2} \) |
| 19 | \( 1 + 7.61T + 19T^{2} \) |
| 23 | \( 1 - 4.16T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 + 2.43T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 - 0.362T + 53T^{2} \) |
| 59 | \( 1 - 0.961T + 59T^{2} \) |
| 61 | \( 1 + 5.32T + 61T^{2} \) |
| 67 | \( 1 - 3.82T + 67T^{2} \) |
| 71 | \( 1 + 5.91T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.59T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 8.00T + 97T^{2} \) |
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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
−8.214201265031443330187430267101, −7.52610185467401195795585323102, −6.59812537969009607466514145365, −6.20664921721096099495042517037, −4.95450553462013457809991164754, −4.40288712440680890765820824318, −3.66453859079484277865774406800, −2.74130273621200033117291433320, −2.05648711657190633701066669675, −0.66838583599836567100472141615,
0.66838583599836567100472141615, 2.05648711657190633701066669675, 2.74130273621200033117291433320, 3.66453859079484277865774406800, 4.40288712440680890765820824318, 4.95450553462013457809991164754, 6.20664921721096099495042517037, 6.59812537969009607466514145365, 7.52610185467401195795585323102, 8.214201265031443330187430267101