L(s) = 1 | + 3·3-s + 14·5-s − 24·7-s + 9·9-s − 28·11-s − 74·13-s + 42·15-s + 82·17-s + 92·19-s − 72·21-s + 8·23-s + 71·25-s + 27·27-s − 138·29-s + 80·31-s − 84·33-s − 336·35-s + 30·37-s − 222·39-s + 282·41-s + 4·43-s + 126·45-s + 240·47-s + 233·49-s + 246·51-s − 130·53-s − 392·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.25·5-s − 1.29·7-s + 1/3·9-s − 0.767·11-s − 1.57·13-s + 0.722·15-s + 1.16·17-s + 1.11·19-s − 0.748·21-s + 0.0725·23-s + 0.567·25-s + 0.192·27-s − 0.883·29-s + 0.463·31-s − 0.443·33-s − 1.62·35-s + 0.133·37-s − 0.911·39-s + 1.07·41-s + 0.0141·43-s + 0.417·45-s + 0.744·47-s + 0.679·49-s + 0.675·51-s − 0.336·53-s − 0.961·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.347376348\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347376348\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 82 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 8 T + p^{3} T^{2} \) |
| 29 | \( 1 + 138 T + p^{3} T^{2} \) |
| 31 | \( 1 - 80 T + p^{3} T^{2} \) |
| 37 | \( 1 - 30 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 130 T + p^{3} T^{2} \) |
| 59 | \( 1 - 596 T + p^{3} T^{2} \) |
| 61 | \( 1 + 218 T + p^{3} T^{2} \) |
| 67 | \( 1 + 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 856 T + p^{3} T^{2} \) |
| 73 | \( 1 + 998 T + p^{3} T^{2} \) |
| 79 | \( 1 + 32 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1508 T + p^{3} T^{2} \) |
| 89 | \( 1 + 246 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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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
−17.20393121536779777605791337554, −16.05096934502075793843671954455, −14.55246735159609396370901763163, −13.45398718290084091243407087898, −12.45955173628784212076292610589, −10.01303079068640580357852296278, −9.538628656859187259243758568889, −7.38704591093503716928355432204, −5.59939847624819033352162579055, −2.78194369232176580575897573097,
2.78194369232176580575897573097, 5.59939847624819033352162579055, 7.38704591093503716928355432204, 9.538628656859187259243758568889, 10.01303079068640580357852296278, 12.45955173628784212076292610589, 13.45398718290084091243407087898, 14.55246735159609396370901763163, 16.05096934502075793843671954455, 17.20393121536779777605791337554