L(s) = 1 | + 23·7-s + 191·13-s + 647·19-s + 625·25-s + 194·31-s + 2.59e3·37-s − 3.21e3·43-s − 1.87e3·49-s − 5.23e3·61-s − 8.80e3·67-s + 9.79e3·73-s − 1.23e4·79-s + 4.39e3·91-s + 9.74e3·97-s + 3.43e3·103-s + 2.20e4·109-s + ⋯ |
L(s) = 1 | + 0.469·7-s + 1.13·13-s + 1.79·19-s + 25-s + 0.201·31-s + 1.89·37-s − 1.73·43-s − 0.779·49-s − 1.40·61-s − 1.96·67-s + 1.83·73-s − 1.98·79-s + 0.530·91-s + 1.03·97-s + 0.323·103-s + 1.85·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.940081404\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.940081404\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( 1 - 23 T + p^{4} T^{2} \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 - 191 T + p^{4} T^{2} \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( 1 - 647 T + p^{4} T^{2} \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 31 | \( 1 - 194 T + p^{4} T^{2} \) |
| 37 | \( 1 - 2591 T + p^{4} T^{2} \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( 1 + 3214 T + p^{4} T^{2} \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 + 5233 T + p^{4} T^{2} \) |
| 67 | \( 1 + 8809 T + p^{4} T^{2} \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 - 9791 T + p^{4} T^{2} \) |
| 79 | \( 1 + 12361 T + p^{4} T^{2} \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 9743 T + p^{4} T^{2} \) |
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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
−13.10150118395261651463374501629, −11.79545170233838888637051890964, −11.01901235611302577516032265134, −9.765795152242794695298418325316, −8.603795071125446036066677679105, −7.51442878040470383663830740308, −6.12385468208258109314980868062, −4.80454257545888379074015510078, −3.21290324573003486762371726122, −1.22535312382696817150276244641,
1.22535312382696817150276244641, 3.21290324573003486762371726122, 4.80454257545888379074015510078, 6.12385468208258109314980868062, 7.51442878040470383663830740308, 8.603795071125446036066677679105, 9.765795152242794695298418325316, 11.01901235611302577516032265134, 11.79545170233838888637051890964, 13.10150118395261651463374501629