| L(s) = 1 | − 234·5-s + 729·9-s + 4.07e3·13-s − 990·17-s + 3.91e4·25-s + 3.18e4·29-s + 5.55e4·37-s − 8.49e4·41-s − 1.70e5·45-s + 1.17e5·49-s + 2.94e4·53-s − 2.34e5·61-s − 9.52e5·65-s + 4.27e5·73-s + 5.31e5·81-s + 2.31e5·85-s + 1.37e6·89-s − 1.47e6·97-s + 1.70e6·101-s − 4.58e5·109-s + 1.12e6·113-s + 2.96e6·117-s + ⋯ |
| L(s) = 1 | − 1.87·5-s + 9-s + 1.85·13-s − 0.201·17-s + 2.50·25-s + 1.30·29-s + 1.09·37-s − 1.23·41-s − 1.87·45-s + 49-s + 0.197·53-s − 1.03·61-s − 3.46·65-s + 1.09·73-s + 81-s + 0.377·85-s + 1.95·89-s − 1.61·97-s + 1.65·101-s − 0.354·109-s + 0.777·113-s + 1.85·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.418131147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418131147\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 + 234 T + p^{6} T^{2} \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 11 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( 1 - 4070 T + p^{6} T^{2} \) |
| 17 | \( 1 + 990 T + p^{6} T^{2} \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 29 | \( 1 - 31878 T + p^{6} T^{2} \) |
| 31 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 37 | \( 1 - 55510 T + p^{6} T^{2} \) |
| 41 | \( 1 + 84942 T + p^{6} T^{2} \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 53 | \( 1 - 29430 T + p^{6} T^{2} \) |
| 59 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 61 | \( 1 + 234938 T + p^{6} T^{2} \) |
| 67 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 71 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 73 | \( 1 - 427570 T + p^{6} T^{2} \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( 1 - 1378962 T + p^{6} T^{2} \) |
| 97 | \( 1 + 1472510 T + p^{6} T^{2} \) |
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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
−13.57934014348252788104404616621, −12.44863570199612483211933887399, −11.46396376625381173503330188564, −10.53216957270590584892298245402, −8.735170337985805936898690563451, −7.80645844956908165125863634403, −6.60260914314922201563084050317, −4.47240918157554411890733309756, −3.52152389120736971652577090206, −0.922560888850850502108824377987,
0.922560888850850502108824377987, 3.52152389120736971652577090206, 4.47240918157554411890733309756, 6.60260914314922201563084050317, 7.80645844956908165125863634403, 8.735170337985805936898690563451, 10.53216957270590584892298245402, 11.46396376625381173503330188564, 12.44863570199612483211933887399, 13.57934014348252788104404616621
