L(s) = 1 | − 2.52·3-s + 3.43·5-s + 7-s + 3.37·9-s − 11-s − 13-s − 8.68·15-s − 6.56·17-s + 2.74·19-s − 2.52·21-s − 2.80·23-s + 6.82·25-s − 0.947·27-s + 0.240·29-s + 0.281·31-s + 2.52·33-s + 3.43·35-s + 5.95·37-s + 2.52·39-s + 11.4·41-s − 5.59·43-s + 11.6·45-s − 13.3·47-s + 49-s + 16.5·51-s + 3.03·53-s − 3.43·55-s + ⋯ |
L(s) = 1 | − 1.45·3-s + 1.53·5-s + 0.377·7-s + 1.12·9-s − 0.301·11-s − 0.277·13-s − 2.24·15-s − 1.59·17-s + 0.629·19-s − 0.550·21-s − 0.584·23-s + 1.36·25-s − 0.182·27-s + 0.0446·29-s + 0.0505·31-s + 0.439·33-s + 0.581·35-s + 0.978·37-s + 0.404·39-s + 1.79·41-s − 0.853·43-s + 1.73·45-s − 1.94·47-s + 0.142·49-s + 2.32·51-s + 0.416·53-s − 0.463·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 5 | \( 1 - 3.43T + 5T^{2} \) |
| 17 | \( 1 + 6.56T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 - 0.240T + 29T^{2} \) |
| 31 | \( 1 - 0.281T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 5.59T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 - 3.03T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 0.183T + 61T^{2} \) |
| 67 | \( 1 - 0.658T + 67T^{2} \) |
| 71 | \( 1 + 5.80T + 71T^{2} \) |
| 73 | \( 1 + 8.78T + 73T^{2} \) |
| 79 | \( 1 + 0.408T + 79T^{2} \) |
| 83 | \( 1 + 5.01T + 83T^{2} \) |
| 89 | \( 1 - 3.72T + 89T^{2} \) |
| 97 | \( 1 - 1.91T + 97T^{2} \) |
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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
−7.25345987483765150148924193563, −6.46155555718940230719085830386, −6.10978380852407147084458744819, −5.49612158987922943698795414067, −4.84634392298320335137198481591, −4.34445902934550497958269136999, −2.86319275703723374856269747016, −2.05663774109585322202331003405, −1.24143660941215913007327243104, 0,
1.24143660941215913007327243104, 2.05663774109585322202331003405, 2.86319275703723374856269747016, 4.34445902934550497958269136999, 4.84634392298320335137198481591, 5.49612158987922943698795414067, 6.10978380852407147084458744819, 6.46155555718940230719085830386, 7.25345987483765150148924193563