L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 11-s + 13-s + 15-s + 2·17-s − 4·19-s − 4·21-s + 25-s + 27-s − 2·29-s + 8·31-s − 33-s − 4·35-s + 6·37-s + 39-s − 6·41-s + 4·43-s + 45-s + 9·49-s + 2·51-s + 2·53-s − 55-s − 4·57-s + 12·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.174·33-s − 0.676·35-s + 0.986·37-s + 0.160·39-s − 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.134·55-s − 0.529·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.417878407\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.417878407\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.38560819787813, −12.92117258888969, −12.74579326384707, −12.14143374599163, −11.51796728762776, −10.98754136636234, −10.20434084228392, −10.04339679866050, −9.731931007736183, −8.986145724179659, −8.694710344767648, −8.127175226721363, −7.551911442641517, −6.915610553183647, −6.529138740323797, −6.039492291812951, −5.582971809330663, −4.847348409485823, −4.153377122242976, −3.718697663912018, −3.038218383729157, −2.650637642248013, −2.088640315501218, −1.197156242911332, −0.4647869098538089,
0.4647869098538089, 1.197156242911332, 2.088640315501218, 2.650637642248013, 3.038218383729157, 3.718697663912018, 4.153377122242976, 4.847348409485823, 5.582971809330663, 6.039492291812951, 6.529138740323797, 6.915610553183647, 7.551911442641517, 8.127175226721363, 8.694710344767648, 8.986145724179659, 9.731931007736183, 10.04339679866050, 10.20434084228392, 10.98754136636234, 11.51796728762776, 12.14143374599163, 12.74579326384707, 12.92117258888969, 13.38560819787813