| L(s) = 1 | + 47·2-s − 1.88e3·4-s + 1.17e5·7-s − 2.81e5·8-s − 3.06e5·11-s + 5.52e6·14-s − 5.48e6·16-s − 1.43e7·22-s − 2.20e8·23-s + 2.44e8·25-s − 2.22e8·28-s + 7.39e8·29-s + 8.93e8·32-s + 5.10e9·37-s + 3.38e9·43-s + 5.78e8·44-s − 1.03e10·46-s + 1.38e10·49-s + 1.14e10·50-s + 4.17e10·53-s − 3.30e10·56-s + 3.47e10·58-s + 6.44e10·64-s − 1.78e11·67-s + 1.97e11·71-s + 2.40e11·74-s − 3.60e10·77-s + ⋯ |
| L(s) = 1 | + 0.734·2-s − 0.460·4-s + 7-s − 1.07·8-s − 0.172·11-s + 0.734·14-s − 0.327·16-s − 0.126·22-s − 1.49·23-s + 25-s − 0.460·28-s + 1.24·29-s + 0.832·32-s + 1.99·37-s + 0.536·43-s + 0.0796·44-s − 1.09·46-s + 49-s + 0.734·50-s + 1.88·53-s − 1.07·56-s + 0.912·58-s + 0.938·64-s − 1.96·67-s + 1.54·71-s + 1.46·74-s − 0.172·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(2.700862036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.700862036\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 - 47 T + p^{12} T^{2} \) |
| 5 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 11 | \( 1 + 306322 T + p^{12} T^{2} \) |
| 13 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 17 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( 1 + 220762978 T + p^{12} T^{2} \) |
| 29 | \( 1 - 739273358 T + p^{12} T^{2} \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( 1 - 5108772818 T + p^{12} T^{2} \) |
| 41 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 43 | \( 1 - 3388378898 T + p^{12} T^{2} \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( 1 - 41794002542 T + p^{12} T^{2} \) |
| 59 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( 1 + 178008750862 T + p^{12} T^{2} \) |
| 71 | \( 1 - 197404987358 T + p^{12} T^{2} \) |
| 73 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 79 | \( 1 - 377568555842 T + p^{12} T^{2} \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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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
−12.51091424896043273347471496414, −11.58614451144468225713669212948, −10.24578684451103891433693101301, −8.871926368288598129983564035511, −7.84889904877213774894606200411, −6.14456712244620399455774560209, −4.96930335354135041905670347632, −4.06321192599137826626861320254, −2.51724001756781562154295924997, −0.821444257569072568774899879578,
0.821444257569072568774899879578, 2.51724001756781562154295924997, 4.06321192599137826626861320254, 4.96930335354135041905670347632, 6.14456712244620399455774560209, 7.84889904877213774894606200411, 8.871926368288598129983564035511, 10.24578684451103891433693101301, 11.58614451144468225713669212948, 12.51091424896043273347471496414
