L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 + 0.707i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·29-s − 2i·31-s + (1.00 + 1.00i)33-s + (−0.707 + 0.707i)35-s + (−0.707 − 0.707i)45-s + 49-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (−0.707 + 0.707i)5-s + 7-s + 1.00i·9-s + 1.41·11-s − 1.00·15-s + (0.707 + 0.707i)21-s − 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·29-s − 2i·31-s + (1.00 + 1.00i)33-s + (−0.707 + 0.707i)35-s + (−0.707 − 0.707i)45-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.717791141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.717791141\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−8.993987651953445974714123409375, −8.178009593016907521912967339886, −7.64369953147051857066927143775, −6.90700298074410231478255790143, −5.95491213842865573657691854722, −4.86799405981738750101015052068, −4.16649903218583304958968876556, −3.61785539223039676946142487638, −2.63614992736551446230516377625, −1.57001496787540789937103265395,
1.10876481567756506747855409122, 1.77887031087284907845848740657, 3.09009880211533391955907447744, 4.02201929751453621718769019674, 4.57323467328878776556738246294, 5.64700887850613080103923232603, 6.61090448548451921108999578610, 7.30017482631736170280606702631, 7.985118324050991841945193991206, 8.611145496268883855267973884394