L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 5·7-s − 8-s + 9-s − 10-s + 5·11-s − 12-s + 3·13-s + 5·14-s − 15-s + 16-s − 18-s − 4·19-s + 20-s + 5·21-s − 5·22-s − 23-s + 24-s + 25-s − 3·26-s − 27-s − 5·28-s + 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.832·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 1.09·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s − 0.944·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−12.56472394654707, −12.19400495171955, −11.96037638616475, −11.26266358938606, −10.80032639763090, −10.36048722510768, −9.972298494275008, −9.674775583015441, −8.996371931104170, −8.814439195245577, −8.459630408897757, −7.523821556059892, −7.030072417110504, −6.589157566758392, −6.400614413683361, −5.928015204738248, −5.630938435536043, −4.658137954850977, −4.109785405254941, −3.682820704387486, −3.144941978321700, −2.550989175532419, −1.907874610752803, −1.165171472538111, −0.7542181904908498, 0,
0.7542181904908498, 1.165171472538111, 1.907874610752803, 2.550989175532419, 3.144941978321700, 3.682820704387486, 4.109785405254941, 4.658137954850977, 5.630938435536043, 5.928015204738248, 6.400614413683361, 6.589157566758392, 7.030072417110504, 7.523821556059892, 8.459630408897757, 8.814439195245577, 8.996371931104170, 9.674775583015441, 9.972298494275008, 10.36048722510768, 10.80032639763090, 11.26266358938606, 11.96037638616475, 12.19400495171955, 12.56472394654707