| L(s) = 1 | + 3-s + 0.693·5-s − 3.39·7-s + 9-s + 5.07·11-s + 3.83·13-s + 0.693·15-s + 3.10·17-s + 6.00·19-s − 3.39·21-s + 1.28·23-s − 4.51·25-s + 27-s + 4.22·29-s − 5.03·31-s + 5.07·33-s − 2.35·35-s − 6.18·37-s + 3.83·39-s + 4.22·41-s + 5.19·43-s + 0.693·45-s − 4.33·47-s + 4.51·49-s + 3.10·51-s − 6.40·53-s + 3.51·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.310·5-s − 1.28·7-s + 0.333·9-s + 1.52·11-s + 1.06·13-s + 0.179·15-s + 0.752·17-s + 1.37·19-s − 0.740·21-s + 0.267·23-s − 0.903·25-s + 0.192·27-s + 0.783·29-s − 0.903·31-s + 0.882·33-s − 0.397·35-s − 1.01·37-s + 0.613·39-s + 0.659·41-s + 0.791·43-s + 0.103·45-s − 0.632·47-s + 0.645·49-s + 0.434·51-s − 0.880·53-s + 0.474·55-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
| \(L(1)\) |
\(\approx\) |
\(2.891240777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891240777\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.693T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 - 5.07T + 11T^{2} \) |
| 13 | \( 1 - 3.83T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 - 6.00T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 - 4.22T + 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 - 4.22T + 41T^{2} \) |
| 43 | \( 1 - 5.19T + 43T^{2} \) |
| 47 | \( 1 + 4.33T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 9.50T + 61T^{2} \) |
| 67 | \( 1 + 7.04T + 67T^{2} \) |
| 71 | \( 1 - 2.91T + 71T^{2} \) |
| 73 | \( 1 - 2.05T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 7.04T + 83T^{2} \) |
| 89 | \( 1 - 1.33T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.097432670959744324302123803312, −7.32147523993559031159947385768, −6.60664402244272140474350250715, −6.09990707049883666105661673301, −5.36024479700772785444763158950, −4.13614042719517885554133228718, −3.48221937148785301677642591722, −3.08977013520440098578772361767, −1.76610006573199754962730717046, −0.927721276384703073467029088074,
0.927721276384703073467029088074, 1.76610006573199754962730717046, 3.08977013520440098578772361767, 3.48221937148785301677642591722, 4.13614042719517885554133228718, 5.36024479700772785444763158950, 6.09990707049883666105661673301, 6.60664402244272140474350250715, 7.32147523993559031159947385768, 8.097432670959744324302123803312
