L(s) = 1 | − 6i·13-s + 8i·17-s − 4·29-s + 2i·37-s + 8·41-s + 7·49-s + 4i·53-s − 10·61-s + 6i·73-s + 16·89-s + 18i·97-s + 20·101-s + 6·109-s − 16i·113-s + ⋯ |
L(s) = 1 | − 1.66i·13-s + 1.94i·17-s − 0.742·29-s + 0.328i·37-s + 1.24·41-s + 49-s + 0.549i·53-s − 1.28·61-s + 0.702i·73-s + 1.69·89-s + 1.82i·97-s + 1.99·101-s + 0.574·109-s − 1.50i·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.781996718\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.781996718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 8iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−7.86525173514319815020497147189, −7.53474560842852664337600832857, −6.41407560446021853875375724891, −5.87860697880945728589687111393, −5.31132024368129011568183282600, −4.30333064255006905260535824483, −3.62377809725290529015974927750, −2.84064616231402384676982044137, −1.84982776907543542952844315641, −0.798601029614064725131281668440,
0.57246951932044064453465913609, 1.81743418591856838576353063873, 2.57168439133104188386623896220, 3.52181995607506869210138467685, 4.39823260994153429687110909318, 4.91183953578439261851188084410, 5.81459721016314760954929307818, 6.53798594357173020496496625948, 7.29472298568322006547671488210, 7.59686529993448976073526515344