| L(s) = 1 | + 1.02e3·2-s + 1.14e5·3-s + 1.04e6·4-s + 1.16e8·6-s + 1.07e9·8-s + 9.56e9·9-s − 4.23e10·11-s + 1.19e11·12-s + 1.09e12·16-s − 3.35e12·17-s + 9.79e12·18-s − 1.01e12·19-s − 4.34e13·22-s + 1.22e14·24-s + 9.53e13·25-s + 6.93e14·27-s + 1.12e15·32-s − 4.84e15·33-s − 3.43e15·34-s + 1.00e16·36-s − 1.03e15·38-s − 2.54e16·41-s + 2.78e15·43-s − 4.44e16·44-s + 1.25e17·48-s + 7.97e16·49-s + 9.76e16·50-s + ⋯ |
| L(s) = 1 | + 2-s + 1.93·3-s + 4-s + 1.93·6-s + 8-s + 2.74·9-s − 1.63·11-s + 1.93·12-s + 16-s − 1.66·17-s + 2.74·18-s − 0.165·19-s − 1.63·22-s + 1.93·24-s + 25-s + 3.36·27-s + 32-s − 3.16·33-s − 1.66·34-s + 2.74·36-s − 0.165·38-s − 1.89·41-s + 0.128·43-s − 1.63·44-s + 1.93·48-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(6.385289577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.385289577\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{10} T \) |
| good | 3 | \( 1 - 114226 T + p^{20} T^{2} \) |
| 5 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 7 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 11 | \( 1 + 42383023726 T + p^{20} T^{2} \) |
| 13 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 17 | \( 1 + 3353535763774 T + p^{20} T^{2} \) |
| 19 | \( 1 + 1014654432526 T + p^{20} T^{2} \) |
| 23 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 29 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 31 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 37 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 41 | \( 1 + 25418071370591326 T + p^{20} T^{2} \) |
| 43 | \( 1 - 2781113986388498 T + p^{20} T^{2} \) |
| 47 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 53 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 59 | \( 1 + 173912197184497198 T + p^{20} T^{2} \) |
| 61 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 67 | \( 1 + 356137514166464974 T + p^{20} T^{2} \) |
| 71 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 73 | \( 1 + 6016717170316692574 T + p^{20} T^{2} \) |
| 79 | \( ( 1 - p^{10} T )( 1 + p^{10} T ) \) |
| 83 | \( 1 + 31022856480301602574 T + p^{20} T^{2} \) |
| 89 | \( 1 - 61202446863210984674 T + p^{20} T^{2} \) |
| 97 | \( 1 + 50009130514058267902 T + p^{20} T^{2} \) |
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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
−15.79940195485861624712976432897, −15.03012238398468829382782711615, −13.64000556972193717297830740544, −12.92389809175974000285833457325, −10.43912973810416712728046572522, −8.505732636221698784273241567536, −7.14688562005276719945333114426, −4.59021675822229857157808834383, −3.03945142575726488506881529048, −2.07479144646133450674174069486,
2.07479144646133450674174069486, 3.03945142575726488506881529048, 4.59021675822229857157808834383, 7.14688562005276719945333114426, 8.505732636221698784273241567536, 10.43912973810416712728046572522, 12.92389809175974000285833457325, 13.64000556972193717297830740544, 15.03012238398468829382782711615, 15.79940195485861624712976432897
