L(s) = 1 | + (−0.707 − 0.707i)2-s + (1.22 − 1.22i)3-s + 1.00i·4-s − 1.73·6-s + (1.50 − 1.50i)7-s + (0.707 − 0.707i)8-s + 5.03i·11-s + (1.22 + 1.22i)12-s + (−3.34 + 3.34i)13-s − 2.12·14-s − 1.00·16-s + (5.06 − 5.06i)17-s + 5.03·19-s − 3.68i·21-s + (3.56 − 3.56i)22-s + (3.80 + 2.91i)23-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (0.707 − 0.707i)3-s + 0.500i·4-s − 0.707·6-s + (0.569 − 0.569i)7-s + (0.250 − 0.250i)8-s + 1.51i·11-s + (0.353 + 0.353i)12-s + (−0.928 + 0.928i)13-s − 0.569·14-s − 0.250·16-s + (1.22 − 1.22i)17-s + 1.15·19-s − 0.804i·21-s + (0.759 − 0.759i)22-s + (0.794 + 0.607i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.767753595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.767753595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-3.80 - 2.91i)T \) |
good | 3 | \( 1 + (-1.22 + 1.22i)T - 3iT^{2} \) |
| 7 | \( 1 + (-1.50 + 1.50i)T - 7iT^{2} \) |
| 11 | \( 1 - 5.03iT - 11T^{2} \) |
| 13 | \( 1 + (3.34 - 3.34i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5.06 + 5.06i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.03T + 19T^{2} \) |
| 29 | \( 1 + 5.46iT - 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + (-4.11 + 4.11i)T - 37iT^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + (1.10 + 1.10i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.34 + 3.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.73 - 9.73i)T + 53iT^{2} \) |
| 59 | \( 1 + 0.535iT - 59T^{2} \) |
| 61 | \( 1 + 3.68iT - 61T^{2} \) |
| 67 | \( 1 + (2.05 - 2.05i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.07T + 71T^{2} \) |
| 73 | \( 1 + (-0.568 + 0.568i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.70T + 79T^{2} \) |
| 83 | \( 1 + (11.3 + 11.3i)T + 83iT^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + (-1.10 + 1.10i)T - 97iT^{2} \) |
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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
−9.772322648440518690668791887803, −8.982488708994489355556867631522, −7.82607837954032922233854371887, −7.38238094555867317241187298855, −7.02424175595216554463162428451, −5.16019756905450950330247129267, −4.45876390958007905437596035652, −3.06984751324514861118445204938, −2.15186161392460824341045600335, −1.21465450102686913086961458047,
1.06604561344271333853877089250, 2.86569030841930079833956018643, 3.49999681627905408041835875843, 5.00313008709541436144797067269, 5.55577315810128863002670399748, 6.58439514412455923829250568282, 7.81231177817166426113499805145, 8.420215307852136102586547577862, 8.806874764246741994230252447597, 9.987800790578319060151781552471