| L(s) = 1 | + 15·4-s − 10·9-s + 24·11-s + 161·16-s − 80·19-s + 100·29-s − 496·31-s − 150·36-s + 804·41-s + 360·44-s − 49·49-s + 1.12e3·59-s − 556·61-s + 1.45e3·64-s + 1.34e3·71-s − 1.20e3·76-s + 2.00e3·79-s − 629·81-s + 1.74e3·89-s − 240·99-s + 964·101-s + 2.34e3·109-s + 1.50e3·116-s − 2.23e3·121-s − 7.44e3·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 0.370·9-s + 0.657·11-s + 2.51·16-s − 0.965·19-s + 0.640·29-s − 2.87·31-s − 0.694·36-s + 3.06·41-s + 1.23·44-s − 1/7·49-s + 2.47·59-s − 1.16·61-s + 2.84·64-s + 2.24·71-s − 1.81·76-s + 2.84·79-s − 0.862·81-s + 2.07·89-s − 0.243·99-s + 0.949·101-s + 2.05·109-s + 1.20·116-s − 1.67·121-s − 5.38·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.876027048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.876027048\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 15 T^{2} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )( 1 + 4 p T + p^{3} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 104 T + p^{3} T^{2} )( 1 + 104 T + p^{3} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 40 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 23310 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 p T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 87050 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 402 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 154390 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 79650 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297270 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 560 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 278 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 574630 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 672 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 771310 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1000 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 942870 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 870 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 772670 T^{2} + p^{6} T^{4} \) |
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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
−12.38597398153129605794497172400, −12.03792772974367796674417379571, −11.32821158822130832436365795288, −10.97926106025997917510025018179, −10.87838899908434603395638930918, −10.17424334118830634099311518650, −9.474333181569463571581048333503, −9.005415380618861998639070929929, −8.350249535703730902348964340949, −7.58796461744719569251968612394, −7.40440256435848423755663086978, −6.64561597158810119852346930793, −6.24940949084594194504415550287, −5.77423258586648356496145048097, −5.05140811523217906291649972598, −3.91361976697285924084149212510, −3.46047683728797660129090803460, −2.37050143334289110838796169626, −2.07345904946879623586996390502, −0.901209372802974164812012261636,
0.901209372802974164812012261636, 2.07345904946879623586996390502, 2.37050143334289110838796169626, 3.46047683728797660129090803460, 3.91361976697285924084149212510, 5.05140811523217906291649972598, 5.77423258586648356496145048097, 6.24940949084594194504415550287, 6.64561597158810119852346930793, 7.40440256435848423755663086978, 7.58796461744719569251968612394, 8.350249535703730902348964340949, 9.005415380618861998639070929929, 9.474333181569463571581048333503, 10.17424334118830634099311518650, 10.87838899908434603395638930918, 10.97926106025997917510025018179, 11.32821158822130832436365795288, 12.03792772974367796674417379571, 12.38597398153129605794497172400
