| L(s) = 1 | + 4-s + 5-s + 2·9-s − 3·16-s + 20-s + 25-s + 4·29-s − 8·31-s + 2·36-s − 4·41-s + 2·45-s + 2·49-s + 4·61-s − 7·64-s − 16·71-s + 8·79-s − 3·80-s − 5·81-s + 4·89-s + 100-s − 4·101-s + 12·109-s + 4·116-s + 121-s − 8·124-s + 125-s + 127-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.447·5-s + 2/3·9-s − 3/4·16-s + 0.223·20-s + 1/5·25-s + 0.742·29-s − 1.43·31-s + 1/3·36-s − 0.624·41-s + 0.298·45-s + 2/7·49-s + 0.512·61-s − 7/8·64-s − 1.89·71-s + 0.900·79-s − 0.335·80-s − 5/9·81-s + 0.423·89-s + 1/10·100-s − 0.398·101-s + 1.14·109-s + 0.371·116-s + 1/11·121-s − 0.718·124-s + 0.0894·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15125 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.307118876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.307118876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( 1 - T \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.94632476461572120916856875406, −10.71956875873175339206765760202, −9.997511322132790519695096125329, −9.614380944770895828298264562601, −8.881480232039856736585626633574, −8.505067314677749635741935550810, −7.51033082953027937199197619340, −7.20447152713543069028854826277, −6.53938732436331094044717882799, −5.98020746457711246784623379348, −5.16995786857303692347586231128, −4.50103395733710191644256088187, −3.63262068636308523102837564548, −2.60602479062541794415271637696, −1.67195028604630473969178666345,
1.67195028604630473969178666345, 2.60602479062541794415271637696, 3.63262068636308523102837564548, 4.50103395733710191644256088187, 5.16995786857303692347586231128, 5.98020746457711246784623379348, 6.53938732436331094044717882799, 7.20447152713543069028854826277, 7.51033082953027937199197619340, 8.505067314677749635741935550810, 8.881480232039856736585626633574, 9.614380944770895828298264562601, 9.997511322132790519695096125329, 10.71956875873175339206765760202, 10.94632476461572120916856875406
