| L(s) = 1 | + 32·7-s + 38·9-s + 36·17-s − 96·23-s − 25·25-s + 352·31-s − 372·41-s + 336·47-s + 82·49-s + 1.21e3·63-s − 336·71-s − 1.01e3·73-s + 544·79-s + 715·81-s + 2.02e3·89-s − 1.53e3·97-s + 896·103-s − 1.11e3·113-s + 1.15e3·119-s − 938·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 1.36e3·153-s + ⋯ |
| L(s) = 1 | + 1.72·7-s + 1.40·9-s + 0.513·17-s − 0.870·23-s − 1/5·25-s + 2.03·31-s − 1.41·41-s + 1.04·47-s + 0.239·49-s + 2.43·63-s − 0.561·71-s − 1.62·73-s + 0.774·79-s + 0.980·81-s + 2.41·89-s − 1.60·97-s + 0.857·103-s − 0.929·113-s + 0.887·119-s − 0.704·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.722·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.892087537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.892087537\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 938 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 3002 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 11782 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 48 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 14182 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 176 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 186 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 149014 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 168 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 49750 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 347254 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 450598 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 471770 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 506 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 272 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 244870 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1014 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 766 T + p^{3} T^{2} )^{2} \) |
| 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
−9.600699533506129862482298384315, −9.044520360984614485699064851009, −8.576778381850420114834659477726, −8.189210238505971206402470548770, −7.943905863717243792594823654605, −7.45514003884469449929755936706, −7.26706757995409809430561704735, −6.54226808307365916686948466102, −6.29321664868313514834996007443, −5.72038944332784288825910446124, −5.02882927787274090627974955843, −4.90343600576618686352199438668, −4.41153223044335345375897833048, −4.01580766187010385167119342148, −3.49761377049754483073197938612, −2.73265252885552569789024311973, −2.15655008097861574747557209962, −1.46604723324269914626221672084, −1.37191594816374592094971124817, −0.52959907677084989457371569709,
0.52959907677084989457371569709, 1.37191594816374592094971124817, 1.46604723324269914626221672084, 2.15655008097861574747557209962, 2.73265252885552569789024311973, 3.49761377049754483073197938612, 4.01580766187010385167119342148, 4.41153223044335345375897833048, 4.90343600576618686352199438668, 5.02882927787274090627974955843, 5.72038944332784288825910446124, 6.29321664868313514834996007443, 6.54226808307365916686948466102, 7.26706757995409809430561704735, 7.45514003884469449929755936706, 7.943905863717243792594823654605, 8.189210238505971206402470548770, 8.576778381850420114834659477726, 9.044520360984614485699064851009, 9.600699533506129862482298384315
