L(s) = 1 | + (−0.421 − 1.34i)2-s + (1.90 + 1.90i)3-s + (−1.64 + 1.13i)4-s + (−0.987 − 2.00i)5-s + (1.77 − 3.38i)6-s + (3.15 − 3.15i)7-s + (2.23 + 1.73i)8-s + 4.28i·9-s + (−2.29 + 2.17i)10-s + 1.76i·11-s + (−5.31 − 0.964i)12-s + (0.707 − 0.707i)13-s + (−5.58 − 2.92i)14-s + (1.94 − 5.71i)15-s + (1.40 − 3.74i)16-s + (2.56 + 2.56i)17-s + ⋯ |
L(s) = 1 | + (−0.298 − 0.954i)2-s + (1.10 + 1.10i)3-s + (−0.822 + 0.569i)4-s + (−0.441 − 0.897i)5-s + (0.723 − 1.38i)6-s + (1.19 − 1.19i)7-s + (0.788 + 0.614i)8-s + 1.42i·9-s + (−0.724 + 0.689i)10-s + 0.533i·11-s + (−1.53 − 0.278i)12-s + (0.196 − 0.196i)13-s + (−1.49 − 0.781i)14-s + (0.502 − 1.47i)15-s + (0.351 − 0.936i)16-s + (0.622 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37244 - 0.531993i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37244 - 0.531993i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.421 + 1.34i)T \) |
| 5 | \( 1 + (0.987 + 2.00i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.90 - 1.90i)T + 3iT^{2} \) |
| 7 | \( 1 + (-3.15 + 3.15i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.76iT - 11T^{2} \) |
| 17 | \( 1 + (-2.56 - 2.56i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 + (2.73 + 2.73i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.75iT - 29T^{2} \) |
| 31 | \( 1 + 4.10iT - 31T^{2} \) |
| 37 | \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + (-0.431 - 0.431i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.54 - 8.54i)T - 47iT^{2} \) |
| 53 | \( 1 + (10.2 - 10.2i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.03T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + (-1.41 + 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (5.51 - 5.51i)T - 73iT^{2} \) |
| 79 | \( 1 - 6.65T + 79T^{2} \) |
| 83 | \( 1 + (-3.92 - 3.92i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.02iT - 89T^{2} \) |
| 97 | \( 1 + (-1.20 - 1.20i)T + 97iT^{2} \) |
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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
−11.68614061413785850421830875466, −10.69152010830608949451383272350, −10.02157198246318726414785577175, −9.118999574433749660302985399474, −8.096978126077074616165203544941, −7.81924167276493466215860618245, −4.87455758614626488825450475909, −4.32556552315419593551865206882, −3.41993995366238740625201755326, −1.55016955213805251086938340694,
1.84026423655313947734310356150, 3.33716351902619654907321475775, 5.21549234746010210038384158722, 6.42793310381778700295628777320, 7.40606583807430180266604158188, 8.167664768497557218213881760228, 8.616760692432569086267291489897, 9.825979734095312981341206502468, 11.33574201413479424997696144040, 12.11233069962182770196775843575