| L(s) = 1 | + 3i·2-s + 3-s − 4-s + 3i·6-s − 15i·7-s + 21i·8-s − 26·9-s − 48i·11-s − 12-s + (−26 − 39i)13-s + 45·14-s − 71·16-s + 45·17-s − 78i·18-s − 6i·19-s + ⋯ |
| L(s) = 1 | + 1.06i·2-s + 0.192·3-s − 0.125·4-s + 0.204i·6-s − 0.809i·7-s + 0.928i·8-s − 0.962·9-s − 1.31i·11-s − 0.0240·12-s + (−0.554 − 0.832i)13-s + 0.859·14-s − 1.10·16-s + 0.642·17-s − 1.02i·18-s − 0.0724i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.088279752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.088279752\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + (26 + 39i)T \) |
| good | 2 | \( 1 - 3iT - 8T^{2} \) |
| 3 | \( 1 - T + 27T^{2} \) |
| 7 | \( 1 + 15iT - 343T^{2} \) |
| 11 | \( 1 + 48iT - 1.33e3T^{2} \) |
| 17 | \( 1 - 45T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 162T + 1.21e4T^{2} \) |
| 29 | \( 1 + 144T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 303iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 192iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 97T + 7.95e4T^{2} \) |
| 47 | \( 1 + 111iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 414T + 1.48e5T^{2} \) |
| 59 | \( 1 - 522iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 376T + 2.26e5T^{2} \) |
| 67 | \( 1 + 36iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 357iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.09e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 830T + 4.93e5T^{2} \) |
| 83 | \( 1 + 438iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 438iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 852iT - 9.12e5T^{2} \) |
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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
−11.08901852382505023402800342386, −10.07368893299840293320815981297, −8.776109462494075077971767080295, −7.947411013749239470553087911918, −7.37662572910284568030856995385, −5.91917980331755965356246432853, −5.60480823720799333696488765638, −3.86726533457738107730464561535, −2.55389193969546803947076989347, −0.34896079559050533861708036371,
1.79613056946970582335240530010, 2.59127335187426901867464451999, 3.85046817371009230904399438447, 5.18943802108346095698281155288, 6.46204188243484931426598993140, 7.56063547925826965968456805760, 8.819927898941978049548062138942, 9.654690206605123634191325950075, 10.37018441619501395731664029039, 11.62006941168649852457873177696
