| L(s) = 1 | + 136·13-s − 50·25-s − 136·37-s − 572·49-s + 1.96e3·61-s − 1.32e3·73-s + 880·97-s + 7.63e3·109-s − 620·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.77e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.90·13-s − 2/5·25-s − 0.604·37-s − 1.66·49-s + 4.12·61-s − 2.12·73-s + 0.921·97-s + 6.71·109-s − 0.465·121-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 + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.26·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.32070895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.32070895\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2}( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 310 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 9817 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 565 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 15259 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 32902 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 27755 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 34 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 47842 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 56458 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 1714 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 266425 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 185530 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 491 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 p T + p^{3} T^{2} )^{2}( 1 + 16 p T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 318334 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 332 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 568691 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 426211 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 1078162 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 220 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.09613940057447487734595168147, −5.74787454892961606393451294179, −5.65618280374295592788796641461, −5.59925207376098885779685272937, −5.49788852837141822992749053261, −4.82631638637749438830547276805, −4.73949415897019639968202398453, −4.61630544035872720539905142596, −4.55621522498966696102235900688, −3.98778587672961202349029275840, −3.75438954430623237107791584446, −3.72619475149358971151461828209, −3.55737749527986171978653466804, −3.31127117298581055638806351747, −3.08809456885638032217718731853, −2.68297158684374484961690943871, −2.57326942431656596688161863464, −2.10776770179782581120239491950, −1.77104423473474554280871243872, −1.68669400458992056809699039025, −1.56980052703751923465001275090, −0.981143262707105799552505719427, −0.78287514180235501315856590759, −0.59887525882974462965664434984, −0.32974355411708018458060083078,
0.32974355411708018458060083078, 0.59887525882974462965664434984, 0.78287514180235501315856590759, 0.981143262707105799552505719427, 1.56980052703751923465001275090, 1.68669400458992056809699039025, 1.77104423473474554280871243872, 2.10776770179782581120239491950, 2.57326942431656596688161863464, 2.68297158684374484961690943871, 3.08809456885638032217718731853, 3.31127117298581055638806351747, 3.55737749527986171978653466804, 3.72619475149358971151461828209, 3.75438954430623237107791584446, 3.98778587672961202349029275840, 4.55621522498966696102235900688, 4.61630544035872720539905142596, 4.73949415897019639968202398453, 4.82631638637749438830547276805, 5.49788852837141822992749053261, 5.59925207376098885779685272937, 5.65618280374295592788796641461, 5.74787454892961606393451294179, 6.09613940057447487734595168147
