L(s) = 1 | + (0.190 + 0.587i)2-s + (0.5 − 0.363i)4-s − 0.618i·7-s + (0.809 + 0.587i)8-s + (0.951 − 0.309i)11-s + (0.363 − 0.118i)14-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.363 + 0.5i)22-s + (0.309 + 0.951i)23-s + (−0.224 − 0.309i)28-s + (−0.951 − 1.30i)29-s + 32-s + (−0.190 + 0.587i)34-s + (−1.53 − 0.5i)37-s + (0.190 − 0.587i)38-s + ⋯ |
L(s) = 1 | + (0.190 + 0.587i)2-s + (0.5 − 0.363i)4-s − 0.618i·7-s + (0.809 + 0.587i)8-s + (0.951 − 0.309i)11-s + (0.363 − 0.118i)14-s + (0.809 + 0.587i)17-s + (−0.809 − 0.587i)19-s + (0.363 + 0.5i)22-s + (0.309 + 0.951i)23-s + (−0.224 − 0.309i)28-s + (−0.951 − 1.30i)29-s + 32-s + (−0.190 + 0.587i)34-s + (−1.53 − 0.5i)37-s + (0.190 − 0.587i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.814753259\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.814753259\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + 0.618iT - T^{2} \) |
| 11 | \( 1 + (-0.951 + 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−8.694963801476350562316918180464, −7.82376882045222861586065657412, −7.28372089157090954529625760987, −6.51463640767243350767248234583, −5.95486752473156564757581521308, −5.17573704514736112174493575727, −4.19562795238856637335805730037, −3.47778989169615192267272742265, −2.15155011242515581392475843215, −1.18712356863815812952797011067,
1.43160450307850129520232372989, 2.25252821613182854163250819550, 3.23193908112851736057354896403, 3.89023645500129368226229263369, 4.84288962609552941080537534787, 5.75104921929179846135189755015, 6.69968513588744557734505972401, 7.13815426204121128206874913953, 8.097557626405148496907089449777, 8.815986022271907191301223247750