L(s) = 1 | + (0.728 + 0.684i)3-s + (0.309 − 0.951i)4-s + (−0.456 + 0.718i)7-s + (0.0627 + 0.998i)9-s + (0.876 − 0.481i)12-s + (−0.0534 + 0.113i)13-s + (−0.809 − 0.587i)16-s + (0.598 − 1.84i)19-s + (−0.824 + 0.211i)21-s + (−0.992 + 0.125i)25-s + (−0.637 + 0.770i)27-s + (0.542 + 0.656i)28-s + (−0.996 − 0.394i)31-s + (0.968 + 0.248i)36-s + (−0.620 + 1.31i)37-s + ⋯ |
L(s) = 1 | + (0.728 + 0.684i)3-s + (0.309 − 0.951i)4-s + (−0.456 + 0.718i)7-s + (0.0627 + 0.998i)9-s + (0.876 − 0.481i)12-s + (−0.0534 + 0.113i)13-s + (−0.809 − 0.587i)16-s + (0.598 − 1.84i)19-s + (−0.824 + 0.211i)21-s + (−0.992 + 0.125i)25-s + (−0.637 + 0.770i)27-s + (0.542 + 0.656i)28-s + (−0.996 − 0.394i)31-s + (0.968 + 0.248i)36-s + (−0.620 + 1.31i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.066278483\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066278483\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.728 - 0.684i)T \) |
| 151 | \( 1 + (0.809 + 0.587i)T \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 7 | \( 1 + (0.456 - 0.718i)T + (-0.425 - 0.904i)T^{2} \) |
| 11 | \( 1 + (-0.0627 + 0.998i)T^{2} \) |
| 13 | \( 1 + (0.0534 - 0.113i)T + (-0.637 - 0.770i)T^{2} \) |
| 17 | \( 1 + (0.425 - 0.904i)T^{2} \) |
| 19 | \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 31 | \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \) |
| 37 | \( 1 + (0.620 - 1.31i)T + (-0.637 - 0.770i)T^{2} \) |
| 41 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 43 | \( 1 + (0.200 + 0.316i)T + (-0.425 + 0.904i)T^{2} \) |
| 47 | \( 1 + (-0.968 - 0.248i)T^{2} \) |
| 53 | \( 1 + (-0.535 - 0.844i)T^{2} \) |
| 59 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \) |
| 67 | \( 1 + (-0.110 + 1.74i)T + (-0.992 - 0.125i)T^{2} \) |
| 71 | \( 1 + (0.425 + 0.904i)T^{2} \) |
| 73 | \( 1 + (-0.331 + 0.521i)T + (-0.425 - 0.904i)T^{2} \) |
| 79 | \( 1 + (-1.84 + 0.730i)T + (0.728 - 0.684i)T^{2} \) |
| 83 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 89 | \( 1 + (-0.876 + 0.481i)T^{2} \) |
| 97 | \( 1 + (0.101 - 1.61i)T + (-0.992 - 0.125i)T^{2} \) |
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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
−11.14401461571433489091220440315, −10.34423103336024855168342324453, −9.300586644394961918126260858905, −9.151200829855189725843786181703, −7.72640629099365215180881110821, −6.63755741078755329947012676709, −5.51847677602389967890194713002, −4.69371561750947074136715658477, −3.18792114001954230880093534067, −2.11402466544849394990064177644,
1.91681145185990929842250150677, 3.35831716457718182325110235821, 3.91804109694894800022395346417, 5.86611889024054448614352351783, 6.97306241726755056638597553891, 7.61047176316991741749802719700, 8.300186357801880117083129615732, 9.355820746478680650244983550902, 10.28558023621108807430934838062, 11.47878223549252250960134459909