| L(s) = 1 | − 6·7-s − 34·13-s − 22·19-s + 18·25-s − 100·31-s − 66·37-s − 20·43-s − 71·49-s − 82·61-s − 166·67-s + 254·73-s − 38·79-s + 204·91-s + 334·97-s + 106·103-s + 20·109-s + 210·121-s + 127-s + 131-s + 132·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 6/7·7-s − 2.61·13-s − 1.15·19-s + 0.719·25-s − 3.22·31-s − 1.78·37-s − 0.465·43-s − 1.44·49-s − 1.34·61-s − 2.47·67-s + 3.47·73-s − 0.481·79-s + 2.24·91-s + 3.44·97-s + 1.02·103-s + 0.183·109-s + 1.73·121-s + 0.00787·127-s + 0.00763·131-s + 0.992·133-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186624 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1424781938\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1424781938\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 18 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 210 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 222 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 510 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 530 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 50 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 33 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2210 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2782 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5490 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6162 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 41 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 83 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 9570 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 127 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 19 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1710 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 8642 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 167 T + p^{2} T^{2} )^{2} \) |
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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
−11.12718998245210481197635504263, −10.54906348475663350918342334161, −10.33391738609716560972992298690, −9.799866147441622006760498669093, −9.360060231517895628417094323807, −9.000955228634846692522942584013, −8.623376043019676415054753426677, −7.64919093060692383585907435905, −7.57123370564936611551740957039, −6.97139127663726731865472956504, −6.62155383368022938788128818209, −6.00754829809896821333307836839, −5.31244935373609583981281313242, −4.90480434718935487675992344962, −4.48319828457366031250193232475, −3.43511976705639095334560767054, −3.28117342469814515435069926464, −2.21902624292853729033189333976, −1.88752505165128720355875051796, −0.14777695675130279212497312687,
0.14777695675130279212497312687, 1.88752505165128720355875051796, 2.21902624292853729033189333976, 3.28117342469814515435069926464, 3.43511976705639095334560767054, 4.48319828457366031250193232475, 4.90480434718935487675992344962, 5.31244935373609583981281313242, 6.00754829809896821333307836839, 6.62155383368022938788128818209, 6.97139127663726731865472956504, 7.57123370564936611551740957039, 7.64919093060692383585907435905, 8.623376043019676415054753426677, 9.000955228634846692522942584013, 9.360060231517895628417094323807, 9.799866147441622006760498669093, 10.33391738609716560972992298690, 10.54906348475663350918342334161, 11.12718998245210481197635504263
