L(s) = 1 | + (0.707 − 0.707i)2-s + (0.352 + 1.69i)3-s − 1.00i·4-s + (−0.499 + 0.499i)5-s + (1.44 + 0.949i)6-s + (1.39 − 1.39i)7-s + (−0.707 − 0.707i)8-s + (−2.75 + 1.19i)9-s + 0.705i·10-s + (−3.39 − 3.39i)11-s + (1.69 − 0.352i)12-s + (−2.39 + 2.69i)13-s − 1.97i·14-s + (−1.02 − 0.670i)15-s − 1.00·16-s + 4.38·17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.203 + 0.979i)3-s − 0.500i·4-s + (−0.223 + 0.223i)5-s + (0.591 + 0.387i)6-s + (0.528 − 0.528i)7-s + (−0.250 − 0.250i)8-s + (−0.916 + 0.398i)9-s + 0.223i·10-s + (−1.02 − 1.02i)11-s + (0.489 − 0.101i)12-s + (−0.665 + 0.746i)13-s − 0.528i·14-s + (−0.263 − 0.173i)15-s − 0.250·16-s + 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.18274 - 0.0183909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.18274 - 0.0183909i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-0.352 - 1.69i)T \) |
| 13 | \( 1 + (2.39 - 2.69i)T \) |
good | 5 | \( 1 + (0.499 - 0.499i)T - 5iT^{2} \) |
| 7 | \( 1 + (-1.39 + 1.39i)T - 7iT^{2} \) |
| 11 | \( 1 + (3.39 + 3.39i)T + 11iT^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + (1.70 + 1.70i)T + 19iT^{2} \) |
| 23 | \( 1 + 0.998T + 23T^{2} \) |
| 29 | \( 1 - 0.998iT - 29T^{2} \) |
| 31 | \( 1 + (-6.50 - 6.50i)T + 31iT^{2} \) |
| 37 | \( 1 + (-4.10 + 4.10i)T - 37iT^{2} \) |
| 41 | \( 1 + (-5.24 + 5.24i)T - 41iT^{2} \) |
| 43 | \( 1 + 8.88iT - 43T^{2} \) |
| 47 | \( 1 + (0.352 + 0.352i)T + 47iT^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 + (-0.998 - 0.998i)T + 59iT^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 + (5.79 + 5.79i)T + 67iT^{2} \) |
| 71 | \( 1 + (7.13 - 7.13i)T - 71iT^{2} \) |
| 73 | \( 1 + (1.70 - 1.70i)T - 73iT^{2} \) |
| 79 | \( 1 + 0.207T + 79T^{2} \) |
| 83 | \( 1 + (-9.17 + 9.17i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.54 - 2.54i)T + 89iT^{2} \) |
| 97 | \( 1 + (3.58 + 3.58i)T + 97iT^{2} \) |
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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
−14.30142785525974245584870728705, −13.69544548474860964886954928868, −12.11403955135732743447431686375, −10.96414730470624087704116206867, −10.39245232860030462040959725117, −9.017779496211895769090585892608, −7.62242054668809914822230571460, −5.59306071394779687634853653645, −4.39666394941560438579021476071, −2.98354879085718255505348241214,
2.58061358722273534316177629584, 4.85920092912969207655615249622, 6.12280122949724851640943093104, 7.77166933207939510687873159385, 8.070512401414089130228438668706, 9.928290778793127880461919992862, 11.73956778549319446338487541750, 12.49037879460743243834997966432, 13.25910100604355932545658689119, 14.63203265175365226667393628914