| L(s) = 1 | + 1.41·5-s + 1.41·13-s − 2·17-s − 2.99·25-s − 4.24·29-s − 7.07·37-s + 10·41-s − 7·49-s − 12.7·53-s + 1.41·61-s + 2.00·65-s − 6·73-s − 2.82·85-s − 16·89-s − 8·97-s + 12.7·101-s + 9.89·109-s − 16·113-s + ⋯ |
| L(s) = 1 | + 0.632·5-s + 0.392·13-s − 0.485·17-s − 0.599·25-s − 0.787·29-s − 1.16·37-s + 1.56·41-s − 49-s − 1.74·53-s + 0.181·61-s + 0.248·65-s − 0.702·73-s − 0.306·85-s − 1.69·89-s − 0.812·97-s + 1.26·101-s + 0.948·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 7.07T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.37685703496192901044300065441, −6.60911702221311962985694789449, −6.00572283757213578165633461735, −5.42489545218306605610735904734, −4.59917277350848073226637324373, −3.84743609344081809515585761392, −3.01445867919703423135729861720, −2.10163795549860412406988230960, −1.38807573819246094102559431375, 0,
1.38807573819246094102559431375, 2.10163795549860412406988230960, 3.01445867919703423135729861720, 3.84743609344081809515585761392, 4.59917277350848073226637324373, 5.42489545218306605610735904734, 6.00572283757213578165633461735, 6.60911702221311962985694789449, 7.37685703496192901044300065441
