| L(s) = 1 | − 1.02e3·2-s + 2.19e4·3-s + 1.04e6·4-s − 2.24e7·6-s + 7.22e8·7-s − 1.07e9·8-s − 9.97e9·9-s + 4.99e10·11-s + 2.29e10·12-s + 2.43e10·13-s − 7.40e11·14-s + 1.09e12·16-s + 5.76e12·17-s + 1.02e13·18-s − 3.02e13·19-s + 1.58e13·21-s − 5.11e13·22-s + 1.45e14·23-s − 2.35e13·24-s − 2.49e13·26-s − 4.48e14·27-s + 7.57e14·28-s − 1.16e15·29-s − 7.43e15·31-s − 1.12e15·32-s + 1.09e15·33-s − 5.90e15·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.214·3-s + 1/2·4-s − 0.151·6-s + 0.967·7-s − 0.353·8-s − 0.954·9-s + 0.580·11-s + 0.107·12-s + 0.0489·13-s − 0.683·14-s + 1/4·16-s + 0.694·17-s + 0.674·18-s − 1.13·19-s + 0.207·21-s − 0.410·22-s + 0.732·23-s − 0.0757·24-s − 0.0346·26-s − 0.418·27-s + 0.483·28-s − 0.515·29-s − 1.62·31-s − 0.176·32-s + 0.124·33-s − 0.490·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{10} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 812 p^{3} T + p^{21} T^{2} \) |
| 7 | \( 1 - 103250464 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 49976398572 T + p^{21} T^{2} \) |
| 13 | \( 1 - 24351400354 T + p^{21} T^{2} \) |
| 17 | \( 1 - 5768874283278 T + p^{21} T^{2} \) |
| 19 | \( 1 + 1592117156980 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 145464074718144 T + p^{21} T^{2} \) |
| 29 | \( 1 + 1167107530943250 T + p^{21} T^{2} \) |
| 31 | \( 1 + 7431907384909648 T + p^{21} T^{2} \) |
| 37 | \( 1 - 54621676523378698 T + p^{21} T^{2} \) |
| 41 | \( 1 + 20889593910177078 T + p^{21} T^{2} \) |
| 43 | \( 1 + 76193896658948996 T + p^{21} T^{2} \) |
| 47 | \( 1 + 508989722403999912 T + p^{21} T^{2} \) |
| 53 | \( 1 + 1341490659556639206 T + p^{21} T^{2} \) |
| 59 | \( 1 - 2306403035584927500 T + p^{21} T^{2} \) |
| 61 | \( 1 - 5268341017122878702 T + p^{21} T^{2} \) |
| 67 | \( 1 - 10904975405120900788 T + p^{21} T^{2} \) |
| 71 | \( 1 + 13367810994741254088 T + p^{21} T^{2} \) |
| 73 | \( 1 - 17952853611107729014 T + p^{21} T^{2} \) |
| 79 | \( 1 - \)\(11\!\cdots\!40\)\( T + p^{21} T^{2} \) |
| 83 | \( 1 + \)\(13\!\cdots\!16\)\( T + p^{21} T^{2} \) |
| 89 | \( 1 - \)\(34\!\cdots\!90\)\( T + p^{21} T^{2} \) |
| 97 | \( 1 - \)\(37\!\cdots\!98\)\( T + p^{21} T^{2} \) |
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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
−10.97065971588653934960032301664, −9.509503135606735138233572089737, −8.543706230159154504207538056054, −7.73397392737915263369070246662, −6.35801488376210377119452342004, −5.12542386658079277278195790180, −3.60789956646067310426533791050, −2.28390635595601231059595751905, −1.27832684333797492153453630323, 0,
1.27832684333797492153453630323, 2.28390635595601231059595751905, 3.60789956646067310426533791050, 5.12542386658079277278195790180, 6.35801488376210377119452342004, 7.73397392737915263369070246662, 8.543706230159154504207538056054, 9.509503135606735138233572089737, 10.97065971588653934960032301664
