| L(s) = 1 | + 176·9-s − 640·25-s + 1.17e3·49-s + 1.64e4·81-s − 4.80e3·113-s + 2.07e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.56e4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s − 1.12e5·225-s + 227-s + ⋯ |
| L(s) = 1 | + 6.51·9-s − 5.11·25-s + 3.41·49-s + 22.5·81-s − 3.99·113-s + 1.55·121-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 − 7.10·169-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 + 0.000300·223-s − 33.3·225-s + 0.000292·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.718321424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.718321424\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 - 586 T^{2} + p^{6} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 44 T^{2} + p^{6} T^{4} )^{4} \) |
| 5 | \( ( 1 + 32 p T^{2} + p^{6} T^{4} )^{4} \) |
| 11 | \( ( 1 - 518 T^{2} + p^{6} T^{4} )^{4} \) |
| 13 | \( ( 1 + 3904 T^{2} + p^{6} T^{4} )^{4} \) |
| 17 | \( ( 1 - 8554 T^{2} + p^{6} T^{4} )^{4} \) |
| 19 | \( ( 1 - 28 T^{2} + p^{6} T^{4} )^{4} \) |
| 23 | \( ( 1 - 9934 T^{2} + p^{6} T^{4} )^{4} \) |
| 29 | \( ( 1 + 2102 T^{2} + p^{6} T^{4} )^{4} \) |
| 31 | \( ( 1 + 27782 T^{2} + p^{6} T^{4} )^{4} \) |
| 37 | \( ( 1 - 88586 T^{2} + p^{6} T^{4} )^{4} \) |
| 41 | \( ( 1 - 10642 T^{2} + p^{6} T^{4} )^{4} \) |
| 43 | \( ( 1 + 3194 T^{2} + p^{6} T^{4} )^{4} \) |
| 47 | \( ( 1 + 53734 T^{2} + p^{6} T^{4} )^{4} \) |
| 53 | \( ( 1 - 1778 p T^{2} + p^{6} T^{4} )^{4} \) |
| 59 | \( ( 1 - 363148 T^{2} + p^{6} T^{4} )^{4} \) |
| 61 | \( ( 1 - 206528 T^{2} + p^{6} T^{4} )^{4} \) |
| 67 | \( ( 1 + 572906 T^{2} + p^{6} T^{4} )^{4} \) |
| 71 | \( ( 1 - 649258 T^{2} + p^{6} T^{4} )^{4} \) |
| 73 | \( ( 1 + 149254 T^{2} + p^{6} T^{4} )^{4} \) |
| 79 | \( ( 1 - 985402 T^{2} + p^{6} T^{4} )^{4} \) |
| 83 | \( ( 1 - 76 p^{2} T^{2} + p^{6} T^{4} )^{4} \) |
| 89 | \( ( 1 - 614938 T^{2} + p^{6} T^{4} )^{4} \) |
| 97 | \( ( 1 - 1824074 T^{2} + p^{6} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.18817551905877410864521275843, −4.17986974849053206335341205524, −4.08368055304948436106881156823, −3.93444354309413272695439055853, −3.92147884926257790573889425650, −3.82827714657619757157095586549, −3.70319543030689617304870304481, −3.55403926604365699933743029977, −3.50376549658693848633055686754, −3.19952351059588188637486629347, −2.68784808557814932279694897965, −2.45550367425356957365136662733, −2.43672935245179078139078775287, −2.40855818416100971580629714365, −2.23754924592727862199012043768, −1.84478438921143817624291799346, −1.84222107434467434833607170089, −1.57664332183217219807588132907, −1.30103110972431511064041727692, −1.29260488720197810485904421636, −1.25279823919948117965955848469, −1.11879738256907411954763047179, −0.61715673959250618943300437904, −0.41182911990036954701637883935, −0.14318150706716931564479461240,
0.14318150706716931564479461240, 0.41182911990036954701637883935, 0.61715673959250618943300437904, 1.11879738256907411954763047179, 1.25279823919948117965955848469, 1.29260488720197810485904421636, 1.30103110972431511064041727692, 1.57664332183217219807588132907, 1.84222107434467434833607170089, 1.84478438921143817624291799346, 2.23754924592727862199012043768, 2.40855818416100971580629714365, 2.43672935245179078139078775287, 2.45550367425356957365136662733, 2.68784808557814932279694897965, 3.19952351059588188637486629347, 3.50376549658693848633055686754, 3.55403926604365699933743029977, 3.70319543030689617304870304481, 3.82827714657619757157095586549, 3.92147884926257790573889425650, 3.93444354309413272695439055853, 4.08368055304948436106881156823, 4.17986974849053206335341205524, 4.18817551905877410864521275843
Plot not available for L-functions of degree greater than 10.
