| L(s) = 1 | − 4.03e3·7-s + 3.58e4·13-s − 5.17e5·19-s − 3.90e5·25-s + 1.80e6·31-s + 1.00e6·37-s − 3.49e6·43-s + 5.76e6·49-s + 2.38e7·61-s + 5.42e6·67-s + 3.23e7·73-s + 1.88e7·79-s − 1.44e8·91-s − 1.76e8·97-s − 4.44e7·103-s + 4.06e8·109-s − 2.14e8·121-s + 127-s + 131-s + 2.08e9·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.68·7-s + 1.25·13-s − 3.96·19-s − 25-s + 1.95·31-s + 0.537·37-s − 1.02·43-s + 49-s + 1.72·61-s + 0.269·67-s + 1.13·73-s + 0.484·79-s − 2.10·91-s − 1.99·97-s − 0.394·103-s + 2.87·109-s − 121-s + 6.66·133-s + 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2240867905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2240867905\) |
| \(L(5)\) |
|
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 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 239 T + p^{8} T^{2} )( 1 + 4273 T + p^{8} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 56447 T + p^{8} T^{2} )( 1 + 20641 T + p^{8} T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 258526 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 1225967 T + p^{8} T^{2} )( 1 - 583439 T + p^{8} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 503522 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 3344879 T + p^{8} T^{2} )( 1 + 6837073 T + p^{8} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 24133919 T + p^{8} T^{2} )( 1 + 307393 T + p^{8} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 37296239 T + p^{8} T^{2} )( 1 + 31874833 T + p^{8} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16169282 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 74894159 T + p^{8} T^{2} )( 1 + 56007121 T + p^{8} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p^{4} T + p^{8} T^{2} )( 1 + p^{4} T + p^{8} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 82132513 T + p^{8} T^{2} )( 1 + 94775521 T + p^{8} T^{2} ) \) |
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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
−10.87659784576660504178985328767, −9.970639523467609981163765214133, −9.723228351268854373709354098227, −8.925777560684071260822647622830, −8.572922088094647786347454738559, −8.306940146030782519466177975796, −7.75469643550362731630167330637, −6.65822614359063744180628316725, −6.65203404376633550482421383891, −6.22715716114645624361080937818, −5.95636965011914742950809370120, −5.01241724588277672009945684442, −4.36156935697647920845673272182, −3.77105350827286726438428774383, −3.70005543371782148132541209702, −2.61724699338215760435855471286, −2.41460745194497708422375330832, −1.64287966455978582990305820425, −0.837357573385050783306262478861, −0.11582544503056237028526641737,
0.11582544503056237028526641737, 0.837357573385050783306262478861, 1.64287966455978582990305820425, 2.41460745194497708422375330832, 2.61724699338215760435855471286, 3.70005543371782148132541209702, 3.77105350827286726438428774383, 4.36156935697647920845673272182, 5.01241724588277672009945684442, 5.95636965011914742950809370120, 6.22715716114645624361080937818, 6.65203404376633550482421383891, 6.65822614359063744180628316725, 7.75469643550362731630167330637, 8.306940146030782519466177975796, 8.572922088094647786347454738559, 8.925777560684071260822647622830, 9.723228351268854373709354098227, 9.970639523467609981163765214133, 10.87659784576660504178985328767
