L(s) = 1 | − 0.613i·3-s − 2.23·5-s − 2.64i·7-s + 2.62·9-s + 5.55i·11-s + 1.06·13-s + 1.37i·15-s − 5.75·17-s − 1.62·21-s + 5.00·25-s − 3.45i·27-s − 9.62·29-s + 3.40·33-s + 5.91i·35-s − 0.652i·39-s + ⋯ |
L(s) = 1 | − 0.354i·3-s − 0.999·5-s − 0.999i·7-s + 0.874·9-s + 1.67i·11-s + 0.294·13-s + 0.354i·15-s − 1.39·17-s − 0.354·21-s + 1.00·25-s − 0.664i·27-s − 1.78·29-s + 0.593·33-s + 0.999i·35-s − 0.104i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7786642098\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7786642098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 0.613iT - 3T^{2} \) |
| 11 | \( 1 - 5.55iT - 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 + 5.75T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 - 9.74iT - 79T^{2} \) |
| 83 | \( 1 - 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19.3T + 97T^{2} \) |
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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
−9.352772732985893543405551440115, −8.219922751496078916983212929702, −7.47925380406841560858540693747, −7.10534922443687349148249311043, −6.48580221811321164835965682510, −4.96027897523411492194083412884, −4.24049068364168777709788665729, −3.83927517456956992741399837900, −2.30055851495701987458672385697, −1.23484645323034467890196917071,
0.28784622818536308819919780910, 1.93497822160919932889858279781, 3.23518706687415579094348172684, 3.83857834770084983194978225423, 4.78024269885192141624202915714, 5.65329805451712424615338879573, 6.49199302696941049004786368363, 7.33773644622475018274537531047, 8.266964275535183611635011241872, 8.804522171571868093702367257149