L(s) = 1 | + (0.707 − 0.707i)5-s − 1.41i·17-s + (−1 − i)19-s − 1.41i·23-s − 1.00i·25-s + 2i·31-s + 1.41·47-s + 49-s + (−1 + i)61-s + (1.41 − 1.41i)83-s + (−1.00 − 1.00i)85-s − 1.41·95-s + (−1.41 − 1.41i)107-s + (−1 + i)109-s + 1.41i·113-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)5-s − 1.41i·17-s + (−1 − i)19-s − 1.41i·23-s − 1.00i·25-s + 2i·31-s + 1.41·47-s + 49-s + (−1 + i)61-s + (1.41 − 1.41i)83-s + (−1.00 − 1.00i)85-s − 1.41·95-s + (−1.41 − 1.41i)107-s + (−1 + i)109-s + 1.41i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.310878319\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.310878319\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - iT^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + 1.41iT - T^{2} \) |
| 19 | \( 1 + (1 + i)T + iT^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - iT^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - iT^{2} \) |
| 61 | \( 1 + (1 - i)T - iT^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.002002686987767710152921981063, −8.262908124224959206616384241941, −7.15192696503075006925019399237, −6.60234971271697216687655335946, −5.68055679337971805153941263022, −4.86091543188863848502853288307, −4.37257270182577770138075548419, −2.93846512937755499593440128631, −2.19221258495500574812432076487, −0.839016424498970648298131400376,
1.63171645262505728566897755724, 2.38403496372455170583436097078, 3.59110018690935537894170936780, 4.19027664739008875950954898275, 5.61236514571412499887662629268, 5.93496742182520550803236134885, 6.72587336608474389909056665781, 7.65793195591859326882625928615, 8.226706491707944231056039170170, 9.270750760470153030519876132129