| L(s) = 1 | + 2-s + 4-s − 4.76·7-s + 8-s − 0.960·11-s − 2.24·13-s − 4.76·14-s + 16-s − 0.249·17-s + 19-s − 0.960·22-s − 9.01·23-s − 2.24·26-s − 4.76·28-s − 6.24·29-s + 2.96·31-s + 32-s − 0.249·34-s + 0.0399·37-s + 38-s + 4.96·43-s − 0.960·44-s − 9.01·46-s + 9.49·47-s + 15.7·49-s − 2.24·52-s − 6.84·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.80·7-s + 0.353·8-s − 0.289·11-s − 0.623·13-s − 1.27·14-s + 0.250·16-s − 0.0605·17-s + 0.229·19-s − 0.204·22-s − 1.88·23-s − 0.441·26-s − 0.901·28-s − 1.16·29-s + 0.531·31-s + 0.176·32-s − 0.0428·34-s + 0.00656·37-s + 0.162·38-s + 0.756·43-s − 0.144·44-s − 1.32·46-s + 1.38·47-s + 2.24·49-s − 0.311·52-s − 0.940·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.737863081\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.737863081\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.76T + 7T^{2} \) |
| 11 | \( 1 + 0.960T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 0.249T + 17T^{2} \) |
| 23 | \( 1 + 9.01T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 0.0399T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 - 9.49T + 47T^{2} \) |
| 53 | \( 1 + 6.84T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 + 5.72T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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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
−7.57668643226396160447116285205, −6.96734194945107848409040579507, −6.28722161843378607534484918352, −5.79204253632257805217041888646, −5.09197316285553652745093095958, −3.96695942809167238035922419254, −3.69069985167752475854001438757, −2.68561167526379193647703290328, −2.15341740744582934154275570692, −0.53927409296349439025322965091,
0.53927409296349439025322965091, 2.15341740744582934154275570692, 2.68561167526379193647703290328, 3.69069985167752475854001438757, 3.96695942809167238035922419254, 5.09197316285553652745093095958, 5.79204253632257805217041888646, 6.28722161843378607534484918352, 6.96734194945107848409040579507, 7.57668643226396160447116285205
