L(s) = 1 | + (−0.733 − 0.680i)4-s + (0.365 − 0.930i)7-s + (1.78 − 0.268i)13-s + (0.0747 + 0.997i)16-s + 1.24·19-s + (−0.988 − 0.149i)25-s + (−0.900 + 0.433i)28-s + (0.222 − 0.385i)31-s + (0.0990 + 0.433i)37-s + (0.0931 + 1.24i)43-s + (−0.733 − 0.680i)49-s + (−1.48 − 1.01i)52-s + (0.326 − 0.302i)61-s + (0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯ |
L(s) = 1 | + (−0.733 − 0.680i)4-s + (0.365 − 0.930i)7-s + (1.78 − 0.268i)13-s + (0.0747 + 0.997i)16-s + 1.24·19-s + (−0.988 − 0.149i)25-s + (−0.900 + 0.433i)28-s + (0.222 − 0.385i)31-s + (0.0990 + 0.433i)37-s + (0.0931 + 1.24i)43-s + (−0.733 − 0.680i)49-s + (−1.48 − 1.01i)52-s + (0.326 − 0.302i)61-s + (0.623 − 0.781i)64-s + (0.900 − 1.56i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.237799413\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.237799413\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.365 + 0.930i)T \) |
good | 2 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 5 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (-1.78 + 0.268i)T + (0.955 - 0.294i)T^{2} \) |
| 17 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 19 | \( 1 - 1.24T + T^{2} \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 31 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \) |
| 41 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 43 | \( 1 + (-0.0931 - 1.24i)T + (-0.988 + 0.149i)T^{2} \) |
| 47 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 53 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 61 | \( 1 + (-0.326 + 0.302i)T + (0.0747 - 0.997i)T^{2} \) |
| 67 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.955 - 0.294i)T^{2} \) |
| 89 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.270385049880378154259073791687, −8.071332634717375423689571977746, −6.99227226181449774061062507395, −6.13804324884952061934921950205, −5.57248441383486000517774798443, −4.66855715763926550077561139109, −3.96719535640406228186660667554, −3.26594953187528141755590027440, −1.63459635893050743701129688073, −0.880365475227726408055824942494,
1.26436646816641966034534676599, 2.50376343285210121369425928558, 3.52722451505704525091784874152, 4.02591663494532379999581680278, 5.15313061888178695616281581547, 5.64038815365887569856144216235, 6.52856194217137909101193339955, 7.53970517195239016174495817557, 8.138514873786352935908576713569, 8.825392044833283806768090182111