| L(s) = 1 | − 4-s + 5-s + 11-s + 16-s − 20-s + 25-s − 2·31-s − 44-s − 49-s + 55-s − 2·59-s − 64-s − 2·71-s + 80-s + 2·89-s − 100-s + ⋯ |
| L(s) = 1 | − 4-s + 5-s + 11-s + 16-s − 20-s + 25-s − 2·31-s − 44-s − 49-s + 55-s − 2·59-s − 64-s − 2·71-s + 80-s + 2·89-s − 100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8635624020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8635624020\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−11.01988493048556701225088240093, −10.09720738134640373280782345645, −9.251794577328399022742427181964, −8.870917978698213797740070258977, −7.58621170924383846889334183869, −6.38988058960110232298233011553, −5.52595649339668195351421013632, −4.52550883304515028228969771566, −3.36164413607351183796270401208, −1.61362900893756487681012342928,
1.61362900893756487681012342928, 3.36164413607351183796270401208, 4.52550883304515028228969771566, 5.52595649339668195351421013632, 6.38988058960110232298233011553, 7.58621170924383846889334183869, 8.870917978698213797740070258977, 9.251794577328399022742427181964, 10.09720738134640373280782345645, 11.01988493048556701225088240093
