L(s) = 1 | + (−0.0581 + 0.998i)3-s + (0.707 − 1.64i)7-s + (−0.993 − 0.116i)9-s + (0.606 − 1.40i)13-s + (0.310 − 1.76i)19-s + (1.59 + 0.802i)21-s + (−0.286 + 0.957i)25-s + (0.173 − 0.984i)27-s + (0.606 + 1.40i)31-s + (−1.18 + 1.59i)37-s + (1.36 + 0.687i)39-s + (−0.439 − 0.368i)43-s + (−1.50 − 1.59i)49-s + (1.73 + 0.412i)57-s + (0.569 + 1.90i)61-s + ⋯ |
L(s) = 1 | + (−0.0581 + 0.998i)3-s + (0.707 − 1.64i)7-s + (−0.993 − 0.116i)9-s + (0.606 − 1.40i)13-s + (0.310 − 1.76i)19-s + (1.59 + 0.802i)21-s + (−0.286 + 0.957i)25-s + (0.173 − 0.984i)27-s + (0.606 + 1.40i)31-s + (−1.18 + 1.59i)37-s + (1.36 + 0.687i)39-s + (−0.439 − 0.368i)43-s + (−1.50 − 1.59i)49-s + (1.73 + 0.412i)57-s + (0.569 + 1.90i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.097019511\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097019511\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0581 - 0.998i)T \) |
| 109 | \( 1 + (-0.173 - 0.984i)T \) |
good | 5 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 7 | \( 1 + (-0.707 + 1.64i)T + (-0.686 - 0.727i)T^{2} \) |
| 11 | \( 1 + (0.993 - 0.116i)T^{2} \) |
| 13 | \( 1 + (-0.606 + 1.40i)T + (-0.686 - 0.727i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 19 | \( 1 + (-0.310 + 1.76i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.396 - 0.918i)T^{2} \) |
| 31 | \( 1 + (-0.606 - 1.40i)T + (-0.686 + 0.727i)T^{2} \) |
| 37 | \( 1 + (1.18 - 1.59i)T + (-0.286 - 0.957i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.439 + 0.368i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.835 + 0.549i)T^{2} \) |
| 53 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 59 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 61 | \( 1 + (-0.569 - 1.90i)T + (-0.835 + 0.549i)T^{2} \) |
| 67 | \( 1 + (0.396 + 0.918i)T + (-0.686 + 0.727i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.0798 - 1.37i)T + (-0.993 + 0.116i)T^{2} \) |
| 79 | \( 1 + (-1.36 - 0.159i)T + (0.973 + 0.230i)T^{2} \) |
| 83 | \( 1 + (-0.597 + 0.802i)T^{2} \) |
| 89 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 97 | \( 1 + (-1.86 + 0.218i)T + (0.973 - 0.230i)T^{2} \) |
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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
−10.18768649460141776537194411994, −8.980727561761258017892742746018, −8.294506119140768667075735011348, −7.42544559992228644626968511513, −6.59078018686425159984525630376, −5.23759102720729083227525200554, −4.81936407251428404110407142160, −3.74013723092460775275951374683, −3.03268360461328837725824986224, −1.04389036731677420392857146296,
1.74595322695743961579296540574, 2.26561269549831114581718344788, 3.71494844944217765935922750663, 5.01153531571986189719688144227, 5.97571309596041082212573738557, 6.31384107469501992007472348987, 7.58157139441529329278526025375, 8.264309523593063433178354707226, 8.841436062587004865718072863530, 9.647891764427749922415376361151