| L(s) = 1 | + (−2 − 2i)3-s + (2 − 2i)7-s + 5i·9-s + (−1 + i)13-s + (5 + 5i)17-s + 4·19-s − 8·21-s + (2 + 2i)23-s + (4 − 4i)27-s + 4i·29-s + 4i·31-s + (1 + i)37-s + 4·39-s + (6 + 6i)43-s + (−2 + 2i)47-s + ⋯ |
| L(s) = 1 | + (−1.15 − 1.15i)3-s + (0.755 − 0.755i)7-s + 1.66i·9-s + (−0.277 + 0.277i)13-s + (1.21 + 1.21i)17-s + 0.917·19-s − 1.74·21-s + (0.417 + 0.417i)23-s + (0.769 − 0.769i)27-s + 0.742i·29-s + 0.718i·31-s + (0.164 + 0.164i)37-s + 0.640·39-s + (0.914 + 0.914i)43-s + (−0.291 + 0.291i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.236908878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.236908878\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (2 + 2i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2 + 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-2 - 2i)T + 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + (-1 - i)T + 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + (2 - 2i)T - 47iT^{2} \) |
| 53 | \( 1 + (7 - 7i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + (-10 + 10i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + (-2 - 2i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−9.413622223939550624798789734255, −8.133125465690917920606161970208, −7.61727826073966844802561949689, −6.99354812874300665672723799699, −6.10907772588770603036871078024, −5.37475069642031071734892128175, −4.58476895883111918494663699182, −3.29613539383094573066654059563, −1.64414702522611555776394123864, −1.04196385229980683940419055957,
0.74224336133570552787174303818, 2.54172334288598827486348885621, 3.71702910636198709638027100023, 4.75438873219656222026554312231, 5.37128852542097409721740056601, 5.76375537033056747436714548833, 7.00621352936811647541120539422, 7.910183342301211870629676942020, 8.879497359957271322294128083006, 9.824035746050276995318001059359
