L(s) = 1 | − 55.9·5-s + 245. i·7-s − 243·9-s + 802i·11-s + 245.·13-s + 1.91e3i·19-s − 3.33e3i·23-s + 3.12e3·25-s − 1.37e4i·35-s − 1.58e4·37-s + 1.52e4·41-s + 1.35e4·45-s + 1.53e4i·47-s − 4.36e4·49-s − 1.68e4·53-s + ⋯ |
L(s) = 1 | − 0.999·5-s + 1.89i·7-s − 9-s + 1.99i·11-s + 0.403·13-s + 1.21i·19-s − 1.31i·23-s + 25-s − 1.89i·35-s − 1.90·37-s + 1.41·41-s + 0.999·45-s + 1.01i·47-s − 2.59·49-s − 0.823·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.6121463614\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6121463614\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 55.9T \) |
good | 3 | \( 1 + 243T^{2} \) |
| 7 | \( 1 - 245. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 802iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 245.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.91e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.33e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 2.05e7T^{2} \) |
| 31 | \( 1 + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.58e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.53e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.68e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.79e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 8.44e8T^{2} \) |
| 67 | \( 1 + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.58e9T^{2} \) |
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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
−11.63257071448113926146192372711, −10.52787881901438152501765879441, −9.292820825268561576898158261374, −8.570027047550695215927031553190, −7.76183234829800881270918482446, −6.48234610938060760380199711343, −5.44310181677161799662068178105, −4.40927904992655342356013119282, −2.97390815804244769131848872014, −1.94739011034032066857403606604,
0.20895829852626990355225425014, 0.883094249850789211345297601727, 3.27258461960101866368017691321, 3.72183536243922442378323533308, 5.11322377232845712621327229034, 6.41801082014443549506739662067, 7.42923494378033084855033595962, 8.235202846878292366075490308459, 9.071612021358279652302542260646, 10.65152090296737630634840641836