L(s) = 1 | + (0.414 + 0.414i)5-s + 3.69i·7-s + (−2.93 − 2.93i)11-s + (−2.41 + 2.41i)13-s + 2.82·17-s + (4.46 − 4.46i)19-s − 6.75i·23-s − 4.65i·25-s + (−5.24 + 5.24i)29-s + 3.06·31-s + (−1.53 + 1.53i)35-s + (−6.41 − 6.41i)37-s − 4i·41-s + (−0.765 − 0.765i)43-s + 3.06·47-s + ⋯ |
L(s) = 1 | + (0.185 + 0.185i)5-s + 1.39i·7-s + (−0.883 − 0.883i)11-s + (−0.669 + 0.669i)13-s + 0.685·17-s + (1.02 − 1.02i)19-s − 1.40i·23-s − 0.931i·25-s + (−0.973 + 0.973i)29-s + 0.549·31-s + (−0.258 + 0.258i)35-s + (−1.05 − 1.05i)37-s − 0.624i·41-s + (−0.116 − 0.116i)43-s + 0.446·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.457748501\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.457748501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.414 - 0.414i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.69iT - 7T^{2} \) |
| 11 | \( 1 + (2.93 + 2.93i)T + 11iT^{2} \) |
| 13 | \( 1 + (2.41 - 2.41i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \) |
| 23 | \( 1 + 6.75iT - 23T^{2} \) |
| 29 | \( 1 + (5.24 - 5.24i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 + (6.41 + 6.41i)T + 37iT^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 + (0.765 + 0.765i)T + 43iT^{2} \) |
| 47 | \( 1 - 3.06T + 47T^{2} \) |
| 53 | \( 1 + (3.24 + 3.24i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.765 - 0.765i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.757 - 0.757i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.39 + 1.39i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.02iT - 71T^{2} \) |
| 73 | \( 1 - 6.48iT - 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + (-9.68 + 9.68i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.82iT - 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−8.395597927170313703683724479530, −7.48634072572198136504487720619, −6.77685423881540882878880565799, −5.90533556757064326402698331447, −5.33583935878823323905743002728, −4.75162762514076690589420498949, −3.44278370232028399581985600166, −2.66337967503140854416957333739, −2.10184230498567696993783739633, −0.45821407081926405887590847091,
0.981318543708285256240329210519, 1.90537814277342388703578281988, 3.21496414290937257495852961715, 3.74111679233574204489428483054, 4.89436553694580317634883458153, 5.27126335501723462181381389798, 6.19307921365242756731808049135, 7.38303443549967555879104829683, 7.54783085661190731885439108844, 8.031178414424578023153192170173