L(s) = 1 | + (0.562 + 0.562i)2-s + (0.534 − 0.534i)3-s − 1.36i·4-s + (−0.233 − 2.22i)5-s + 0.601·6-s + (−0.567 + 0.567i)7-s + (1.89 − 1.89i)8-s + 2.42i·9-s + (1.11 − 1.38i)10-s + 4.83i·11-s + (−0.730 − 0.730i)12-s + (−3.26 + 3.26i)13-s − 0.638·14-s + (−1.31 − 1.06i)15-s − 0.601·16-s + (1.54 − 1.54i)17-s + ⋯ |
L(s) = 1 | + (0.397 + 0.397i)2-s + (0.308 − 0.308i)3-s − 0.683i·4-s + (−0.104 − 0.994i)5-s + 0.245·6-s + (−0.214 + 0.214i)7-s + (0.669 − 0.669i)8-s + 0.809i·9-s + (0.354 − 0.437i)10-s + 1.45i·11-s + (−0.210 − 0.210i)12-s + (−0.904 + 0.904i)13-s − 0.170·14-s + (−0.339 − 0.274i)15-s − 0.150·16-s + (0.375 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30454 - 0.247623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30454 - 0.247623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.233 + 2.22i)T \) |
| 23 | \( 1 + (2.45 + 4.11i)T \) |
good | 2 | \( 1 + (-0.562 - 0.562i)T + 2iT^{2} \) |
| 3 | \( 1 + (-0.534 + 0.534i)T - 3iT^{2} \) |
| 7 | \( 1 + (0.567 - 0.567i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.83iT - 11T^{2} \) |
| 13 | \( 1 + (3.26 - 3.26i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.54 + 1.54i)T - 17iT^{2} \) |
| 19 | \( 1 - 7.21T + 19T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 + 1.58T + 31T^{2} \) |
| 37 | \( 1 + (6.03 - 6.03i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + (4.49 + 4.49i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.427 - 0.427i)T + 47iT^{2} \) |
| 53 | \( 1 + (6.85 + 6.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 7.76iT - 59T^{2} \) |
| 61 | \( 1 - 7.62iT - 61T^{2} \) |
| 67 | \( 1 + (-1.20 + 1.20i)T - 67iT^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + (-0.559 + 0.559i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.66T + 79T^{2} \) |
| 83 | \( 1 + (2.15 + 2.15i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.48T + 89T^{2} \) |
| 97 | \( 1 + (-5.03 + 5.03i)T - 97iT^{2} \) |
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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
−13.69564508028140547519502727514, −12.61548943875293458307722146156, −11.77604066201361668890081541924, −10.00515237095619691564553873530, −9.429219472715463309231356735697, −7.84347119330584374057523577814, −6.91135527999743459857227174057, −5.25875597380419593790207028299, −4.56738454801756714733206321779, −1.91345111599671797539715026534,
3.14992461662657701232646384212, 3.51606841080895166037933950084, 5.56555596448381546002921792285, 7.14943867803600015270418582350, 8.123841751783293507144010919974, 9.533112716036470992484521750431, 10.66503933141376466596335469521, 11.62198528178484861294560137977, 12.51367372226587829835193599724, 13.78452118014866251921718309558