L(s) = 1 | + i·2-s − 1.71i·3-s − 4-s + 1.71·6-s − 2.77i·7-s − i·8-s + 0.0597·9-s + 2.77·11-s + 1.71i·12-s − 5.67i·13-s + 2.77·14-s + 16-s + 5.15i·17-s + 0.0597i·18-s − 1.41·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.989i·3-s − 0.5·4-s + 0.700·6-s − 1.04i·7-s − 0.353i·8-s + 0.0199·9-s + 0.836·11-s + 0.494i·12-s − 1.57i·13-s + 0.741·14-s + 0.250·16-s + 1.25i·17-s + 0.0140i·18-s − 0.324·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.414486442\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.414486442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 1.71iT - 3T^{2} \) |
| 7 | \( 1 + 2.77iT - 7T^{2} \) |
| 11 | \( 1 - 2.77T + 11T^{2} \) |
| 13 | \( 1 + 5.67iT - 13T^{2} \) |
| 17 | \( 1 - 5.15iT - 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 - 0.654iT - 23T^{2} \) |
| 29 | \( 1 + 4.09T + 29T^{2} \) |
| 31 | \( 1 + 7.12T + 31T^{2} \) |
| 37 | \( 1 + 1.04iT - 37T^{2} \) |
| 41 | \( 1 - 9.10T + 41T^{2} \) |
| 43 | \( 1 + 9.24iT - 43T^{2} \) |
| 47 | \( 1 + 2.77iT - 47T^{2} \) |
| 53 | \( 1 - 0.526iT - 53T^{2} \) |
| 59 | \( 1 - 3.78T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.32iT - 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 4.21iT - 73T^{2} \) |
| 79 | \( 1 - 9.90T + 79T^{2} \) |
| 83 | \( 1 + 4.67iT - 83T^{2} \) |
| 89 | \( 1 - 9.18T + 89T^{2} \) |
| 97 | \( 1 + 0.0901iT - 97T^{2} \) |
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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
−9.325285565006339124812941592233, −8.388892359628008126859954333775, −7.57090988344671226074886271537, −7.24043255798475137072821247574, −6.26157424126979701198951015532, −5.62214596598812042076271989179, −4.25858760930924513482000585894, −3.52840428302495339065615874657, −1.77431918345264034419577831693, −0.62609648221076953288456069148,
1.65083638701624411416418529535, 2.75771456122971824493645057370, 3.93125732392760524096456050091, 4.51929570161944660912108432159, 5.43248165906939625474511835233, 6.47969735726945272755119086390, 7.50374686488451969481193907448, 8.914239423337844761166898902983, 9.257224807485805585253036076957, 9.615299807221483827978479614636