L(s) = 1 | + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + 5-s + 8-s − 9-s + 10-s − 2·13-s + 16-s − 2·17-s − 18-s + 20-s + 25-s − 2·26-s + 32-s − 2·34-s − 36-s + 37-s + 40-s + 2·41-s − 45-s − 49-s + 50-s − 2·52-s + 64-s − 2·65-s − 2·68-s − 72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.770809771\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770809771\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( ( 1 + T )^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )^{2} \) |
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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
−10.80821644897585403714162290343, −9.780213099441346580515854375829, −9.026933799111392309276303617302, −7.77567985936988259385724288307, −6.79294902490755101631249068117, −6.06124431521073214267295899939, −5.13488793212626901170948348056, −4.41645981063411314247634537539, −2.72816018386619166937329311600, −2.24576785584387077945318324787,
2.24576785584387077945318324787, 2.72816018386619166937329311600, 4.41645981063411314247634537539, 5.13488793212626901170948348056, 6.06124431521073214267295899939, 6.79294902490755101631249068117, 7.77567985936988259385724288307, 9.026933799111392309276303617302, 9.780213099441346580515854375829, 10.80821644897585403714162290343