L(s) = 1 | + (−0.173 + 0.984i)3-s + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)13-s + (−0.266 − 0.223i)15-s + 1.53·17-s + (1.76 − 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 0.652·35-s + 0.347·37-s + (−0.766 − 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)3-s + (−0.173 + 0.300i)5-s + (−0.939 − 1.62i)7-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)13-s + (−0.266 − 0.223i)15-s + 1.53·17-s + (1.76 − 0.642i)21-s + (0.439 + 0.761i)25-s + (0.5 − 0.866i)27-s + (−0.5 + 0.866i)31-s + 0.652·35-s + 0.347·37-s + (−0.766 − 0.642i)39-s + (0.766 + 1.32i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9027350377\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9027350377\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 - 0.984i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - 1.53T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.87T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−9.231622239530103967789636195657, −7.989389754691687230081687714829, −7.33668810728480254308415327643, −6.69276897530970450303748510276, −5.89021481091345109066053786379, −4.91891875610524054993050614658, −4.16360597452696556417884068996, −3.52115451161312057333861382525, −2.89344185370798417512906918267, −1.07628345669822305692950463093,
0.63111814297088148890816658048, 2.15790323934057622335037041346, 2.77427477467094565633722405222, 3.63843085583174294992786969762, 5.21154120441001507234473902026, 5.57415475340358738537472973091, 6.19723449876003247921626565217, 7.06557001946012807835454684388, 7.84944806275273809174057428316, 8.429330540639321356516658184878