| L(s) = 1 | + 3-s + 1.86·5-s − 3.57·7-s + 9-s + 1.54·11-s − 1.56·13-s + 1.86·15-s − 8.05·17-s + 5.13·19-s − 3.57·21-s − 7.33·23-s − 1.52·25-s + 27-s + 0.759·29-s + 7.02·31-s + 1.54·33-s − 6.67·35-s + 7.25·37-s − 1.56·39-s + 4.08·41-s − 4.19·43-s + 1.86·45-s + 6.88·47-s + 5.81·49-s − 8.05·51-s + 13.4·53-s + 2.88·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.834·5-s − 1.35·7-s + 0.333·9-s + 0.467·11-s − 0.433·13-s + 0.481·15-s − 1.95·17-s + 1.17·19-s − 0.781·21-s − 1.52·23-s − 0.304·25-s + 0.192·27-s + 0.141·29-s + 1.26·31-s + 0.269·33-s − 1.12·35-s + 1.19·37-s − 0.250·39-s + 0.638·41-s − 0.639·43-s + 0.278·45-s + 1.00·47-s + 0.830·49-s − 1.12·51-s + 1.84·53-s + 0.389·55-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
| \(L(1)\) |
\(\approx\) |
\(2.272519640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272519640\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 + 3.57T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 8.05T + 17T^{2} \) |
| 19 | \( 1 - 5.13T + 19T^{2} \) |
| 23 | \( 1 + 7.33T + 23T^{2} \) |
| 29 | \( 1 - 0.759T + 29T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 - 7.25T + 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 - 6.88T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 - 9.15T + 61T^{2} \) |
| 67 | \( 1 + 0.0152T + 67T^{2} \) |
| 71 | \( 1 - 7.68T + 71T^{2} \) |
| 73 | \( 1 - 5.58T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 0.653T + 97T^{2} \) |
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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
−8.164355236897529248960445299847, −7.25540034654539030644492413268, −6.56474413940856257282071175946, −6.16877851735303821956963525537, −5.29713630309756367848340899377, −4.22070916803339284040506639151, −3.68305136339657217348915755108, −2.49430292851262003825756283144, −2.25887313328604803202358450553, −0.74484971111900307671760886008,
0.74484971111900307671760886008, 2.25887313328604803202358450553, 2.49430292851262003825756283144, 3.68305136339657217348915755108, 4.22070916803339284040506639151, 5.29713630309756367848340899377, 6.16877851735303821956963525537, 6.56474413940856257282071175946, 7.25540034654539030644492413268, 8.164355236897529248960445299847
