L(s) = 1 | − 7-s + 9-s + 2·23-s + 25-s + 49-s − 63-s − 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯ |
L(s) = 1 | − 7-s + 9-s + 2·23-s + 25-s + 49-s − 63-s − 2·71-s + 2·79-s + 81-s − 2·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.136957023\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.136957023\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.325473541400985597549032705450, −8.969404930325293933568052340708, −7.77444638889755251251434305092, −6.94619162637443057695498660915, −6.53532043385071100598650967671, −5.37094050327471962140986341606, −4.53472642496952284617012496364, −3.52359602661071549771480199843, −2.66996088697319804910630081465, −1.16763625025495495608525127318,
1.16763625025495495608525127318, 2.66996088697319804910630081465, 3.52359602661071549771480199843, 4.53472642496952284617012496364, 5.37094050327471962140986341606, 6.53532043385071100598650967671, 6.94619162637443057695498660915, 7.77444638889755251251434305092, 8.969404930325293933568052340708, 9.325473541400985597549032705450