L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 13-s + 16-s + 19-s − 27-s − 2·31-s + 36-s − 37-s − 39-s + 2·43-s − 48-s + 52-s − 57-s + 61-s + 64-s − 67-s + 73-s + 76-s − 79-s + 81-s + 2·93-s + 97-s + 103-s − 108-s + ⋯ |
L(s) = 1 | − 3-s + 4-s + 9-s − 12-s + 13-s + 16-s + 19-s − 27-s − 2·31-s + 36-s − 37-s − 39-s + 2·43-s − 48-s + 52-s − 57-s + 61-s + 64-s − 67-s + 73-s + 76-s − 79-s + 81-s + 2·93-s + 97-s + 103-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.308942441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.308942441\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 + T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−8.727111994675082977950945882167, −7.60152937353908639527236995781, −7.23971827010739363746027909896, −6.42056516249941661123190313874, −5.74493870704949820171809228841, −5.26205053768995842603129959590, −4.03254523850355388223015875648, −3.30444621536861241692475051925, −2.02836076922678343683519002717, −1.11280921225040039207153375699,
1.11280921225040039207153375699, 2.02836076922678343683519002717, 3.30444621536861241692475051925, 4.03254523850355388223015875648, 5.26205053768995842603129959590, 5.74493870704949820171809228841, 6.42056516249941661123190313874, 7.23971827010739363746027909896, 7.60152937353908639527236995781, 8.727111994675082977950945882167