| L(s) = 1 | + 3-s + 3·5-s + 7-s + 9-s + 3·11-s + 5·13-s + 3·15-s − 2·19-s + 21-s + 6·23-s + 4·25-s + 27-s + 6·29-s − 4·31-s + 3·33-s + 3·35-s − 11·37-s + 5·39-s + 12·41-s + 43-s + 3·45-s − 12·47-s + 49-s − 9·53-s + 9·55-s − 2·57-s + 12·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 1.38·13-s + 0.774·15-s − 0.458·19-s + 0.218·21-s + 1.25·23-s + 4/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s + 0.522·33-s + 0.507·35-s − 1.80·37-s + 0.800·39-s + 1.87·41-s + 0.152·43-s + 0.447·45-s − 1.75·47-s + 1/7·49-s − 1.23·53-s + 1.21·55-s − 0.264·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.823195641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.823195641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−13.88166597028988, −13.29489376800835, −12.97708228622411, −12.54313074300515, −11.74938864367057, −11.28707720528303, −10.70691082970742, −10.40341240843145, −9.699366756703184, −9.243549424911817, −8.862000669389117, −8.480857927015711, −7.892099905506796, −7.101193120940400, −6.589178674496443, −6.262662433897969, −5.663589445378642, −5.002676099315589, −4.557124368270858, −3.575292710494813, −3.471255576107863, −2.467590308267863, −1.968146009834587, −1.360264922384071, −0.8568929578056549,
0.8568929578056549, 1.360264922384071, 1.968146009834587, 2.467590308267863, 3.471255576107863, 3.575292710494813, 4.557124368270858, 5.002676099315589, 5.663589445378642, 6.262662433897969, 6.589178674496443, 7.101193120940400, 7.892099905506796, 8.480857927015711, 8.862000669389117, 9.243549424911817, 9.699366756703184, 10.40341240843145, 10.70691082970742, 11.28707720528303, 11.74938864367057, 12.54313074300515, 12.97708228622411, 13.29489376800835, 13.88166597028988
