| L(s) = 1 | + 4-s − 2·5-s + 2·7-s + 4·13-s + 16-s − 2·20-s − 25-s + 2·28-s + 10·29-s − 4·35-s − 14·47-s − 3·49-s + 4·52-s + 20·61-s + 64-s − 8·65-s − 10·67-s + 8·73-s + 10·79-s − 2·80-s − 9·81-s + 32·83-s + 8·91-s − 6·97-s − 100-s + 2·112-s + 10·116-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.894·5-s + 0.755·7-s + 1.10·13-s + 1/4·16-s − 0.447·20-s − 1/5·25-s + 0.377·28-s + 1.85·29-s − 0.676·35-s − 2.04·47-s − 3/7·49-s + 0.554·52-s + 2.56·61-s + 1/8·64-s − 0.992·65-s − 1.22·67-s + 0.936·73-s + 1.12·79-s − 0.223·80-s − 81-s + 3.51·83-s + 0.838·91-s − 0.609·97-s − 0.0999·100-s + 0.188·112-s + 0.928·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.259239911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.259239911\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 48 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.156525613946623728262888995571, −7.84649946323976276134361114249, −7.53936765772774627007250403325, −6.71466039866347727089201108202, −6.54647306280285237450115502412, −6.20037920335371970522684408155, −5.34729932771528919043202523905, −5.08656535503202332258124020431, −4.50043924374032736151528972341, −3.95931764930517258516944137950, −3.51078204746398253243680572308, −2.98702510183565899115103525519, −2.22436034322611110505751496146, −1.54310722971679780919710543645, −0.75999533497345359971401024699,
0.75999533497345359971401024699, 1.54310722971679780919710543645, 2.22436034322611110505751496146, 2.98702510183565899115103525519, 3.51078204746398253243680572308, 3.95931764930517258516944137950, 4.50043924374032736151528972341, 5.08656535503202332258124020431, 5.34729932771528919043202523905, 6.20037920335371970522684408155, 6.54647306280285237450115502412, 6.71466039866347727089201108202, 7.53936765772774627007250403325, 7.84649946323976276134361114249, 8.156525613946623728262888995571
