| L(s) = 1 | − 3·9-s + 10·13-s + 10·25-s + 22·37-s − 28·61-s + 34·73-s + 9·81-s + 28·97-s + 38·109-s − 30·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 49·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 9-s + 2.77·13-s + 2·25-s + 3.61·37-s − 3.58·61-s + 3.97·73-s + 81-s + 2.84·97-s + 3.63·109-s − 2.77·117-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364377441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364377441\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−8.999458648083630397226696163831, −8.722116977686320016117657659884, −8.592411771710152035383310514744, −8.060276956475196258282810850670, −7.60328312655264427851249002265, −7.53830314391727339477886692793, −6.55015926135628570757331071405, −6.32394293916374963552184542085, −6.17467587035086162136486554697, −5.90976277719655730971555710512, −5.13173563226316994185506770823, −4.90644529039755256083338493869, −4.31793359595896290640512826791, −3.88837444724613216657744471022, −3.32759459259491112535463312565, −3.08716652618488231727242178286, −2.53998092191621087533239667446, −1.86549826958650923603681110114, −0.986080289151396894388916181761, −0.829786216541165657964021732799,
0.829786216541165657964021732799, 0.986080289151396894388916181761, 1.86549826958650923603681110114, 2.53998092191621087533239667446, 3.08716652618488231727242178286, 3.32759459259491112535463312565, 3.88837444724613216657744471022, 4.31793359595896290640512826791, 4.90644529039755256083338493869, 5.13173563226316994185506770823, 5.90976277719655730971555710512, 6.17467587035086162136486554697, 6.32394293916374963552184542085, 6.55015926135628570757331071405, 7.53830314391727339477886692793, 7.60328312655264427851249002265, 8.060276956475196258282810850670, 8.592411771710152035383310514744, 8.722116977686320016117657659884, 8.999458648083630397226696163831
