| L(s) = 1 | − 4·4-s − 9·9-s + 24·11-s + 16·16-s − 40·19-s − 60·29-s − 176·31-s + 36·36-s + 84·41-s − 96·44-s + 430·49-s + 1.32e3·59-s − 1.07e3·61-s − 64·64-s + 1.58e3·71-s + 160·76-s + 1.04e3·79-s + 81·81-s − 1.62e3·89-s − 216·99-s − 1.23e3·101-s − 2.38e3·109-s + 240·116-s − 2.23e3·121-s + 704·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 0.657·11-s + 1/4·16-s − 0.482·19-s − 0.384·29-s − 1.01·31-s + 1/6·36-s + 0.319·41-s − 0.328·44-s + 1.25·49-s + 2.91·59-s − 2.25·61-s − 1/8·64-s + 2.64·71-s + 0.241·76-s + 1.48·79-s + 1/9·81-s − 1.92·89-s − 0.219·99-s − 1.21·101-s − 2.09·109-s + 0.192·116-s − 1.67·121-s + 0.509·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.551254202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551254202\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 430 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 6050 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3890 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 88 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 42 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156310 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 198430 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 258550 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 538 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 179930 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 792 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 730510 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 520 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 901510 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 810 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 493630 T^{2} + p^{6} T^{4} \) |
| 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
−12.88935979540794446194687714540, −12.11151908684600446439981304031, −12.07742495508161353372830052660, −11.07629486055602669308441468570, −10.97001335860802327418590291120, −10.28158431250180428078133453845, −9.431225990465431248687162928918, −9.421910594330110596214688032552, −8.625501998755661139369293637728, −8.242080266763142198116972280680, −7.55286991847974058170407434075, −6.89055927304721360609988287331, −6.37498172551979777044192897346, −5.54244122201709576918154188481, −5.20018035181641026023594236260, −4.11737922925898063725803719450, −3.85685864719197492741733155384, −2.80293928994829497994758352887, −1.82200097105262489841901868116, −0.61762565942453821655206538542,
0.61762565942453821655206538542, 1.82200097105262489841901868116, 2.80293928994829497994758352887, 3.85685864719197492741733155384, 4.11737922925898063725803719450, 5.20018035181641026023594236260, 5.54244122201709576918154188481, 6.37498172551979777044192897346, 6.89055927304721360609988287331, 7.55286991847974058170407434075, 8.242080266763142198116972280680, 8.625501998755661139369293637728, 9.421910594330110596214688032552, 9.431225990465431248687162928918, 10.28158431250180428078133453845, 10.97001335860802327418590291120, 11.07629486055602669308441468570, 12.07742495508161353372830052660, 12.11151908684600446439981304031, 12.88935979540794446194687714540
