| L(s) = 1 | + 81·9-s − 240·13-s + 480·17-s − 82·29-s − 1.68e3·37-s + 3.03e3·41-s + 2.40e3·49-s + 5.04e3·53-s + 6.95e3·61-s − 1.05e4·73-s + 6.56e3·81-s + 9.75e3·89-s + 1.87e4·97-s + 1.88e4·101-s + 9.36e3·109-s + 6.72e3·113-s − 1.94e4·117-s + ⋯ |
| L(s) = 1 | + 9-s − 1.42·13-s + 1.66·17-s − 0.0975·29-s − 1.22·37-s + 1.80·41-s + 49-s + 1.79·53-s + 1.86·61-s − 1.98·73-s + 81-s + 1.23·89-s + 1.98·97-s + 1.84·101-s + 0.787·109-s + 0.526·113-s − 1.42·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.131516242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131516242\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 240 T + p^{4} T^{2} \) |
| 17 | \( 1 - 480 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 + 82 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 1680 T + p^{4} T^{2} \) |
| 41 | \( 1 - 3038 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 - 5040 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6958 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 10560 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 - 9758 T + p^{4} T^{2} \) |
| 97 | \( 1 - 18720 T + p^{4} T^{2} \) |
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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
−10.32067267595884044925400194617, −9.972141020144458450616346842356, −8.913304055456548022307354901062, −7.58889768127112117228466269958, −7.18333084875310016326210334983, −5.76643327709784560833425409257, −4.80043300896772009352313791575, −3.65938538276525892296608058860, −2.28368285350121840787360836465, −0.869698807101025030937098344722,
0.869698807101025030937098344722, 2.28368285350121840787360836465, 3.65938538276525892296608058860, 4.80043300896772009352313791575, 5.76643327709784560833425409257, 7.18333084875310016326210334983, 7.58889768127112117228466269958, 8.913304055456548022307354901062, 9.972141020144458450616346842356, 10.32067267595884044925400194617
