L(s) = 1 | + (−1 + 1.41i)3-s + (1.73 + 1.41i)5-s + (1 + 2.44i)7-s + (−1.00 − 2.82i)9-s + 2.82i·11-s + 4·13-s + (−3.73 + 1.03i)15-s + 2.82i·17-s + (−4.46 − 1.03i)21-s + 3.46·23-s + (0.999 + 4.89i)25-s + (5.00 + 1.41i)27-s + 5.65i·29-s − 9.79i·31-s + (−4.00 − 2.82i)33-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.816i)3-s + (0.774 + 0.632i)5-s + (0.377 + 0.925i)7-s + (−0.333 − 0.942i)9-s + 0.852i·11-s + 1.10·13-s + (−0.963 + 0.267i)15-s + 0.685i·17-s + (−0.974 − 0.225i)21-s + 0.722·23-s + (0.199 + 0.979i)25-s + (0.962 + 0.272i)27-s + 1.05i·29-s − 1.75i·31-s + (−0.696 − 0.492i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.713518835\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.713518835\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 - 1.41i)T \) |
| 5 | \( 1 + (-1.73 - 1.41i)T \) |
| 7 | \( 1 + (-1 - 2.44i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 + 9.79iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.89iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 9.79iT - 61T^{2} \) |
| 67 | \( 1 + 4.89iT - 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−9.624297275230063093549820303825, −9.084799392966760506830482950966, −8.238352958708656377753397320486, −7.01895896338357310413708831919, −6.16508960360217832343007898580, −5.66917371693870402910452968101, −4.80859884553317516933557605745, −3.79201205800946477740482840659, −2.73402507758405591985094099117, −1.57356564596574809741445957369,
0.78101735804922216981723828745, 1.46572922768665464353829004887, 2.84270818125192484025098966162, 4.18604653068758034872309553944, 5.17124065935850563394472220571, 5.82056735183562409944732391328, 6.63844157156451038617884054003, 7.36562637058268240833868520472, 8.383335128942816973785734164375, 8.805635858110874036752399276459