| L(s) = 1 | + 8·4-s − 7·9-s + 22·11-s + 48·16-s + 74·31-s − 56·36-s + 176·44-s + 98·49-s − 214·59-s + 256·64-s − 266·71-s − 32·81-s − 194·89-s − 154·99-s + 363·121-s + 592·124-s + 127-s + 131-s + 137-s + 139-s − 336·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯ |
| L(s) = 1 | + 2·4-s − 7/9·9-s + 2·11-s + 3·16-s + 2.38·31-s − 1.55·36-s + 4·44-s + 2·49-s − 3.62·59-s + 4·64-s − 3.74·71-s − 0.395·81-s − 2.17·89-s − 1.55·99-s + 3·121-s + 4.77·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 7/3·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.140031405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.140031405\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 3 | $C_2^2$ | \( 1 + 7 T^{2} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 167 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2113 T^{2} + p^{4} T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 1918 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 718 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 107 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 7753 T^{2} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 133 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 97 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 9793 T^{2} + p^{4} T^{4} \) |
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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
−11.79177693603448769702976386782, −11.62737411939730147349944269459, −10.98918670390420101249117952823, −10.69648386059242525717558876116, −10.07752576977510283717954218905, −9.698076997253849789726788020157, −8.825558388945193175894420325015, −8.698221035463996934627816353112, −7.85013072071112084194087207326, −7.46716735871079512811658757486, −6.84550245497527454405280617778, −6.50180985940506148508946609127, −5.89268037500287046657279965399, −5.83218345298344293033893227129, −4.58795891902087215906258400306, −4.00877688340984973565850450219, −2.93342086060253800989636614379, −2.92009732547528960577531915043, −1.74066615048293645852928406887, −1.14491873678030497525607348057,
1.14491873678030497525607348057, 1.74066615048293645852928406887, 2.92009732547528960577531915043, 2.93342086060253800989636614379, 4.00877688340984973565850450219, 4.58795891902087215906258400306, 5.83218345298344293033893227129, 5.89268037500287046657279965399, 6.50180985940506148508946609127, 6.84550245497527454405280617778, 7.46716735871079512811658757486, 7.85013072071112084194087207326, 8.698221035463996934627816353112, 8.825558388945193175894420325015, 9.698076997253849789726788020157, 10.07752576977510283717954218905, 10.69648386059242525717558876116, 10.98918670390420101249117952823, 11.62737411939730147349944269459, 11.79177693603448769702976386782
