L(s) = 1 | + i·2-s − 1.02i·3-s − 4-s − 1.25i·5-s + 1.02·6-s − i·8-s + 1.94·9-s + 1.25·10-s + (−3.30 + 0.235i)11-s + 1.02i·12-s + 4.08·13-s − 1.29·15-s + 16-s − 3.20·17-s + 1.94i·18-s + 7.62·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.593i·3-s − 0.5·4-s − 0.562i·5-s + 0.419·6-s − 0.353i·8-s + 0.648·9-s + 0.398·10-s + (−0.997 + 0.0710i)11-s + 0.296i·12-s + 1.13·13-s − 0.333·15-s + 0.250·16-s − 0.776·17-s + 0.458i·18-s + 1.75·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.464530790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.464530790\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.30 - 0.235i)T \) |
good | 3 | \( 1 + 1.02iT - 3T^{2} \) |
| 5 | \( 1 + 1.25iT - 5T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 + 3.20T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 + 8.25T + 23T^{2} \) |
| 29 | \( 1 + 3.54iT - 29T^{2} \) |
| 31 | \( 1 + 9.18iT - 31T^{2} \) |
| 37 | \( 1 + 0.308T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 + 7.57iT - 43T^{2} \) |
| 47 | \( 1 + 4.70iT - 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 - 2.73iT - 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 - 3.38T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 - 0.966T + 73T^{2} \) |
| 79 | \( 1 + 15.6iT - 79T^{2} \) |
| 83 | \( 1 - 1.32T + 83T^{2} \) |
| 89 | \( 1 + 10.6iT - 89T^{2} \) |
| 97 | \( 1 - 10.6iT - 97T^{2} \) |
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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
−9.647366459682456052684786673199, −8.772454988961750738128637013514, −7.85798267365011368660530295062, −7.50328331972117181465377693692, −6.35887063345525643484814253171, −5.68825297720326991420161684146, −4.67519751380532698821942713081, −3.74562807610733109296884376719, −2.13439456372299629726527988560, −0.72739394603641398179615028384,
1.40048319495272688002495201231, 2.84915042545349786407406957318, 3.61187301580099564916197658359, 4.61677311155788298639113143263, 5.48984717975362770059939438758, 6.60844971072376704654884117866, 7.61459893678164963772703803383, 8.481105315842264297864494417823, 9.460353918626363037445127202302, 10.06486111737702448578787092779