| L(s) = 1 | − i·3-s + (1.51 + 1.64i)5-s − 0.437i·7-s − 9-s − 5.73·11-s + i·13-s + (1.64 − 1.51i)15-s − 3.98i·17-s + 4.77·19-s − 0.437·21-s − 0.337i·23-s + (−0.437 + 4.98i)25-s + i·27-s − 1.72·29-s + 7.86·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (0.675 + 0.737i)5-s − 0.165i·7-s − 0.333·9-s − 1.72·11-s + 0.277i·13-s + (0.425 − 0.389i)15-s − 0.965i·17-s + 1.09·19-s − 0.0954·21-s − 0.0704i·23-s + (−0.0875 + 0.996i)25-s + 0.192i·27-s − 0.319·29-s + 1.41·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.780101837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.780101837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.51 - 1.64i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 + 0.437iT - 7T^{2} \) |
| 11 | \( 1 + 5.73T + 11T^{2} \) |
| 17 | \( 1 + 3.98iT - 17T^{2} \) |
| 19 | \( 1 - 4.77T + 19T^{2} \) |
| 23 | \( 1 + 0.337iT - 23T^{2} \) |
| 29 | \( 1 + 1.72T + 29T^{2} \) |
| 31 | \( 1 - 7.86T + 31T^{2} \) |
| 37 | \( 1 + 1.66iT - 37T^{2} \) |
| 41 | \( 1 - 2.68T + 41T^{2} \) |
| 43 | \( 1 + 2.27iT - 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 + 14.2iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 2.25T + 61T^{2} \) |
| 67 | \( 1 - 2.17iT - 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 2.55iT - 73T^{2} \) |
| 79 | \( 1 - 2.55T + 79T^{2} \) |
| 83 | \( 1 - 5.41iT - 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.443202789477749616766083232916, −7.69811636653851967230424131591, −7.11206867535802616352124249450, −6.48864990865865976512598471217, −5.37529288251178943504700022431, −5.16430289493607183085288624162, −3.62858584378613462780149607969, −2.68114357402759858069296774183, −2.17113307670652219735586477448, −0.65894092984323975932454051474,
0.979229579975708940818704798558, 2.34103309086372336051706024255, 3.06981546719444478636885936054, 4.29568554992396981841409973803, 5.01931067563408015181603307816, 5.63851229220451340399418674754, 6.20939840183108648972905667128, 7.57284348254593339484228266303, 8.085884210085220994989581422980, 8.823895278658221823485626622553
