L(s) = 1 | + (0.988 − 0.149i)3-s + (0.733 − 0.680i)7-s + (0.955 − 0.294i)9-s + (0.440 + 1.92i)13-s + (−0.733 − 1.26i)19-s + (0.623 − 0.781i)21-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (0.0747 − 0.129i)31-s + (0.123 + 1.64i)37-s + (0.722 + 1.84i)39-s + (−1.19 + 1.49i)43-s + (0.0747 − 0.997i)49-s + (−0.914 − 1.14i)57-s + (−0.134 − 1.79i)61-s + ⋯ |
L(s) = 1 | + (0.988 − 0.149i)3-s + (0.733 − 0.680i)7-s + (0.955 − 0.294i)9-s + (0.440 + 1.92i)13-s + (−0.733 − 1.26i)19-s + (0.623 − 0.781i)21-s + (−0.733 − 0.680i)25-s + (0.900 − 0.433i)27-s + (0.0747 − 0.129i)31-s + (0.123 + 1.64i)37-s + (0.722 + 1.84i)39-s + (−1.19 + 1.49i)43-s + (0.0747 − 0.997i)49-s + (−0.914 − 1.14i)57-s + (−0.134 − 1.79i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.847038169\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847038169\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 + (-0.733 + 0.680i)T \) |
good | 5 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (-0.440 - 1.92i)T + (-0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 19 | \( 1 + (0.733 + 1.26i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 29 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.123 - 1.64i)T + (-0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 43 | \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 53 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 73 | \( 1 + (1.21 + 1.12i)T + (0.0747 + 0.997i)T^{2} \) |
| 79 | \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 97 | \( 1 + 0.445T + T^{2} \) |
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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
−9.083641884477387637027579647788, −8.325921676599316531518593439901, −7.78893025376073581103424529016, −6.71083159602797010802595082754, −6.51216555508543579924185307110, −4.69547087697128113346404175547, −4.42854425976628075060312425763, −3.43064038173837170086769614965, −2.23565466144448216190233748107, −1.45222423205320220948568015029,
1.53535039062202461880369649963, 2.48566991054796659615350535615, 3.44206275780468711881843252013, 4.19098268774099503487518003187, 5.42654567278748456830223587752, 5.80039923465728246133306949183, 7.19415088907652404484937794067, 7.890140882552185563146141604973, 8.425834745660845846115042480342, 8.942229662496635373293976839153