L(s) = 1 | + (0.809 + 0.587i)3-s + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)5-s + (0.309 + 0.951i)9-s + i·12-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 + i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.587 + 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (−1.39 + 0.221i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)3-s + (0.587 + 0.809i)4-s + (−0.951 − 0.309i)5-s + (0.309 + 0.951i)9-s + i·12-s + (−0.587 − 0.809i)15-s + (−0.309 + 0.951i)16-s + (−0.309 − 0.951i)20-s + (1 + i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.951i)27-s + (−0.587 + 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (−1.39 + 0.221i)47-s + (−0.809 + 0.587i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.436957853\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.436957853\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (1.39 - 0.221i)T + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−9.442907020808774020835200208925, −8.655567003840483833082946726629, −8.190292800040344033451663153453, −7.37773190266209550570011825782, −6.88057781506889136015895894942, −5.37715892713181366543485477392, −4.49254381980272329292924061391, −3.57374496719152881985784435603, −3.13828919117285998981495708853, −1.86251019477020712152648342432,
1.03488879542574869347554384729, 2.34095629303386294189897794133, 3.12859673559292412674285287669, 4.16719298543467603491519853404, 5.24843999553029490792212808672, 6.40881917787331710919119513636, 6.97882034810401776541886556228, 7.54441120654552481840052216170, 8.520876058625403629877583049485, 9.054707042082629008026933393506