| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 13-s + 14-s + 16-s + 3·17-s − 7·19-s − 6·23-s − 26-s + 28-s + 3·29-s + 8·31-s + 32-s + 3·34-s + 2·37-s − 7·38-s + 6·41-s + 2·43-s − 6·46-s − 3·47-s + 49-s − 52-s + 9·53-s + 56-s + 3·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 1.60·19-s − 1.25·23-s − 0.196·26-s + 0.188·28-s + 0.557·29-s + 1.43·31-s + 0.176·32-s + 0.514·34-s + 0.328·37-s − 1.13·38-s + 0.937·41-s + 0.304·43-s − 0.884·46-s − 0.437·47-s + 1/7·49-s − 0.138·52-s + 1.23·53-s + 0.133·56-s + 0.393·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.425937452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.425937452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.897027663604322173678949232804, −6.80372035451057240722667487356, −6.27982096164805657207952288763, −5.67358607205667203911547653900, −4.79864064774061289985810246364, −4.29843115281266460124353530081, −3.58107677912591984673243535167, −2.56566243850023025022238641415, −2.00080981543475741545929929484, −0.789541050929285862978630047886,
0.789541050929285862978630047886, 2.00080981543475741545929929484, 2.56566243850023025022238641415, 3.58107677912591984673243535167, 4.29843115281266460124353530081, 4.79864064774061289985810246364, 5.67358607205667203911547653900, 6.27982096164805657207952288763, 6.80372035451057240722667487356, 7.897027663604322173678949232804
