| L(s) = 1 | + 64·4-s − 74·5-s + 1.33e3·11-s + 4.09e3·16-s − 4.73e3·20-s + 1.26e4·23-s − 1.01e4·25-s + 5.60e4·31-s + 8.70e4·37-s + 8.51e4·44-s + 2.06e5·47-s + 1.17e5·49-s − 2.46e5·53-s − 9.84e4·55-s − 1.07e5·59-s + 2.62e5·64-s − 4.28e5·67-s + 3.41e5·71-s − 3.03e5·80-s − 1.39e6·89-s + 8.10e5·92-s − 1.82e6·97-s − 6.49e5·100-s − 8.11e5·103-s − 1.70e6·113-s − 9.37e5·115-s + ⋯ |
| L(s) = 1 | + 4-s − 0.591·5-s + 11-s + 16-s − 0.591·20-s + 1.04·23-s − 0.649·25-s + 1.88·31-s + 1.71·37-s + 44-s + 1.98·47-s + 49-s − 1.65·53-s − 0.591·55-s − 0.524·59-s + 64-s − 1.42·67-s + 0.953·71-s − 0.591·80-s − 1.97·89-s + 1.04·92-s − 1.99·97-s − 0.649·100-s − 0.742·103-s − 1.17·113-s − 0.616·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.400308438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.400308438\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - p^{3} T \) |
| good | 2 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 5 | \( 1 + 74 T + p^{6} T^{2} \) |
| 7 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 13 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 17 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 19 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 23 | \( 1 - 12670 T + p^{6} T^{2} \) |
| 29 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 31 | \( 1 - 56018 T + p^{6} T^{2} \) |
| 37 | \( 1 - 87050 T + p^{6} T^{2} \) |
| 41 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 43 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 47 | \( 1 - 206350 T + p^{6} T^{2} \) |
| 53 | \( 1 + 246890 T + p^{6} T^{2} \) |
| 59 | \( 1 + 107642 T + p^{6} T^{2} \) |
| 61 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 67 | \( 1 + 428470 T + p^{6} T^{2} \) |
| 71 | \( 1 - 341278 T + p^{6} T^{2} \) |
| 73 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 79 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 83 | \( ( 1 - p^{3} T )( 1 + p^{3} T ) \) |
| 89 | \( 1 + 1392338 T + p^{6} T^{2} \) |
| 97 | \( 1 + 1824190 T + p^{6} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.40167533687645930297227479179, −11.66304628726322371440788682030, −10.82456155838734545022737398819, −9.499718614958949716433902202399, −8.111095984178669232556655687724, −7.05397717528977001659685094040, −6.01537274521766257344985695978, −4.22288890786733464311656035998, −2.78942526642248072019523905237, −1.10917210842492402559416399554,
1.10917210842492402559416399554, 2.78942526642248072019523905237, 4.22288890786733464311656035998, 6.01537274521766257344985695978, 7.05397717528977001659685094040, 8.111095984178669232556655687724, 9.499718614958949716433902202399, 10.82456155838734545022737398819, 11.66304628726322371440788682030, 12.40167533687645930297227479179
