L(s) = 1 | − 3.07·3-s + (−1.47 − 1.47i)7-s + 6.45·9-s + (1.20 − 1.20i)11-s + 5.63i·13-s + (−4.22 − 4.22i)17-s + (−3.11 + 3.11i)19-s + (4.54 + 4.54i)21-s + (1.08 − 1.08i)23-s − 10.6·27-s + (−5.32 − 5.32i)29-s + 4.67i·31-s + (−3.71 + 3.71i)33-s + 1.51i·37-s − 17.3i·39-s + ⋯ |
L(s) = 1 | − 1.77·3-s + (−0.558 − 0.558i)7-s + 2.15·9-s + (0.363 − 0.363i)11-s + 1.56i·13-s + (−1.02 − 1.02i)17-s + (−0.715 + 0.715i)19-s + (0.991 + 0.991i)21-s + (0.225 − 0.225i)23-s − 2.04·27-s + (−0.988 − 0.988i)29-s + 0.839i·31-s + (−0.646 + 0.646i)33-s + 0.248i·37-s − 2.77i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6267595484\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6267595484\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 + (1.47 + 1.47i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.20 + 1.20i)T - 11iT^{2} \) |
| 13 | \( 1 - 5.63iT - 13T^{2} \) |
| 17 | \( 1 + (4.22 + 4.22i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.11 - 3.11i)T - 19iT^{2} \) |
| 23 | \( 1 + (-1.08 + 1.08i)T - 23iT^{2} \) |
| 29 | \( 1 + (5.32 + 5.32i)T + 29iT^{2} \) |
| 31 | \( 1 - 4.67iT - 31T^{2} \) |
| 37 | \( 1 - 1.51iT - 37T^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 - 2.42iT - 43T^{2} \) |
| 47 | \( 1 + (-0.827 + 0.827i)T - 47iT^{2} \) |
| 53 | \( 1 - 8.17T + 53T^{2} \) |
| 59 | \( 1 + (-7.78 - 7.78i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.03 - 3.03i)T - 61iT^{2} \) |
| 67 | \( 1 + 2.93iT - 67T^{2} \) |
| 71 | \( 1 - 0.180T + 71T^{2} \) |
| 73 | \( 1 + (-2.19 - 2.19i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 - 9.08T + 89T^{2} \) |
| 97 | \( 1 + (-6.04 - 6.04i)T + 97iT^{2} \) |
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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
−9.544340259631369722946227413104, −8.843216185657242583980660550718, −7.41400586282299095716617592971, −6.67156217006811243483676921948, −6.38662107796999709512172368850, −5.36558780474803810492558353687, −4.44671782617121223361513826496, −3.85489421437326547068799478036, −2.01440168605500928860200094961, −0.62187259577410184893485451694,
0.56778690480777502206755950420, 2.09569278458212059755069393802, 3.60532312704904486097710054709, 4.65023364679230169938877700170, 5.44996764609284023679234377850, 6.10307955521310413407635460116, 6.69342551049255202632908198417, 7.57705710458853630082734006651, 8.718417634495725143710802628234, 9.567392585646089892097858922628