| L(s) = 1 | + 540·4-s − 6.56e3·9-s − 2.92e4·11-s + 2.94e4·16-s − 8.83e5·19-s + 2.09e6·29-s − 1.58e7·31-s − 3.54e6·36-s + 2.65e7·41-s − 1.58e7·44-s + 4.48e7·49-s + 6.40e7·59-s + 2.21e8·61-s − 1.25e8·64-s + 5.53e8·71-s − 4.77e8·76-s − 8.96e8·79-s + 4.30e7·81-s − 3.79e8·89-s + 1.92e8·99-s − 2.63e9·101-s − 3.95e9·109-s + 1.13e9·116-s − 4.07e9·121-s − 8.54e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.05·4-s − 1/3·9-s − 0.603·11-s + 0.112·16-s − 1.55·19-s + 0.551·29-s − 3.07·31-s − 0.351·36-s + 1.46·41-s − 0.636·44-s + 1.11·49-s + 0.688·59-s + 2.04·61-s − 0.936·64-s + 2.58·71-s − 1.64·76-s − 2.58·79-s + 1/9·81-s − 0.641·89-s + 0.201·99-s − 2.51·101-s − 2.68·109-s + 0.581·116-s − 1.72·121-s − 3.24·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.558550709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.558550709\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 135 p^{2} T^{2} + p^{18} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 44851070 T^{2} + p^{18} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 14648 T + p^{9} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 19772133910 T^{2} + p^{18} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 42608307390 T^{2} + p^{18} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 441820 T + p^{9} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1524006503570 T^{2} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1049350 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7910568 T + p^{9} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 180764011793210 T^{2} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 13285562 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 470152980649990 T^{2} + p^{18} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2045781727484190 T^{2} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3277713901593990 T^{2} + p^{18} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 32042120 T + p^{9} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 110664022 T + p^{9} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 40354634616070070 T^{2} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 276679712 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 48034873641925390 T^{2} + p^{18} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 448202760 T + p^{9} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 350347374506432810 T^{2} + p^{18} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 189894930 T + p^{9} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 491963359241209150 T^{2} + p^{18} T^{4} \) |
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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
−12.74594626813386424008245941076, −12.56684008616730797580044102699, −11.71701503042245599847971439042, −11.18118239488921377651088810580, −10.80053147684857431110336739471, −10.44890394265082589090948200058, −9.535362288367991398193201668266, −8.986513306100013007822394697892, −8.300180848530345090116581746499, −7.75327791669520941189934943986, −6.85437863371775176420743866317, −6.80336810253944511128318883854, −5.64962045413972067982749364013, −5.48028155969253773839515635547, −4.27320938169622283223041379902, −3.70580092723797461145831204968, −2.54397704664273585491582810438, −2.34731343852841413103705855525, −1.46165159683835399454857114661, −0.33832489089603146709014820786,
0.33832489089603146709014820786, 1.46165159683835399454857114661, 2.34731343852841413103705855525, 2.54397704664273585491582810438, 3.70580092723797461145831204968, 4.27320938169622283223041379902, 5.48028155969253773839515635547, 5.64962045413972067982749364013, 6.80336810253944511128318883854, 6.85437863371775176420743866317, 7.75327791669520941189934943986, 8.300180848530345090116581746499, 8.986513306100013007822394697892, 9.535362288367991398193201668266, 10.44890394265082589090948200058, 10.80053147684857431110336739471, 11.18118239488921377651088810580, 11.71701503042245599847971439042, 12.56684008616730797580044102699, 12.74594626813386424008245941076
