| L(s) = 1 | + 4-s + 9-s − 11-s + 13-s + 16-s − 2·17-s + 23-s − 31-s + 36-s − 41-s − 43-s − 44-s + 47-s + 49-s + 52-s + 53-s − 59-s + 64-s − 2·67-s − 2·68-s − 79-s + 81-s + 83-s + 92-s + 97-s − 99-s − 101-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 11-s + 13-s + 16-s − 2·17-s + 23-s − 31-s + 36-s − 41-s − 43-s − 44-s + 47-s + 49-s + 52-s + 53-s − 59-s + 64-s − 2·67-s − 2·68-s − 79-s + 81-s + 83-s + 92-s + 97-s − 99-s − 101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.316939882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316939882\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 + T )^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−10.49936255341516225772479009606, −9.202076649400370844889748616979, −8.433443998652338867077557941424, −7.35444290980853675333330807091, −6.87697978870141463366852936553, −5.98076140788929993448198631197, −4.91592270842247149923901685161, −3.83260892980893756693182467265, −2.64917534804443801125375407471, −1.61694374948618120152619996050,
1.61694374948618120152619996050, 2.64917534804443801125375407471, 3.83260892980893756693182467265, 4.91592270842247149923901685161, 5.98076140788929993448198631197, 6.87697978870141463366852936553, 7.35444290980853675333330807091, 8.433443998652338867077557941424, 9.202076649400370844889748616979, 10.49936255341516225772479009606
