| L(s) = 1 | − 0.329·3-s − 4.07·7-s − 2.89·9-s − 3.08·11-s − 6.32·13-s + 0.982·17-s − 7.08·19-s + 1.34·21-s + 23-s + 1.94·27-s − 7.63·29-s − 5.92·31-s + 1.01·33-s + 11.4·37-s + 2.08·39-s − 5.74·41-s + 1.74·43-s + 2.24·47-s + 9.57·49-s − 0.323·51-s + 1.31·53-s + 2.33·57-s + 7.92·59-s + 9.51·61-s + 11.7·63-s − 3.34·67-s − 0.329·69-s + ⋯ |
| L(s) = 1 | − 0.190·3-s − 1.53·7-s − 0.963·9-s − 0.931·11-s − 1.75·13-s + 0.238·17-s − 1.62·19-s + 0.292·21-s + 0.208·23-s + 0.373·27-s − 1.41·29-s − 1.06·31-s + 0.177·33-s + 1.88·37-s + 0.333·39-s − 0.896·41-s + 0.266·43-s + 0.328·47-s + 1.36·49-s − 0.0453·51-s + 0.181·53-s + 0.309·57-s + 1.03·59-s + 1.21·61-s + 1.48·63-s − 0.408·67-s − 0.0396·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1839476470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1839476470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 0.329T + 3T^{2} \) |
| 7 | \( 1 + 4.07T + 7T^{2} \) |
| 11 | \( 1 + 3.08T + 11T^{2} \) |
| 13 | \( 1 + 6.32T + 13T^{2} \) |
| 17 | \( 1 - 0.982T + 17T^{2} \) |
| 19 | \( 1 + 7.08T + 19T^{2} \) |
| 29 | \( 1 + 7.63T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 - 1.31T + 53T^{2} \) |
| 59 | \( 1 - 7.92T + 59T^{2} \) |
| 61 | \( 1 - 9.51T + 61T^{2} \) |
| 67 | \( 1 + 3.34T + 67T^{2} \) |
| 71 | \( 1 + 3.92T + 71T^{2} \) |
| 73 | \( 1 + 1.66T + 73T^{2} \) |
| 79 | \( 1 + 15.9T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.84T + 97T^{2} \) |
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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
−8.370577673779143079998751821943, −7.42431133999694264228796352642, −6.96502749731653308834857677708, −5.97288852591006891113714963706, −5.59274330658274793569746702836, −4.65216715794995784076643216594, −3.68370170832003648321763261569, −2.74026234993164975198499785393, −2.29934629462414001072859987758, −0.21704433587242095601448665795,
0.21704433587242095601448665795, 2.29934629462414001072859987758, 2.74026234993164975198499785393, 3.68370170832003648321763261569, 4.65216715794995784076643216594, 5.59274330658274793569746702836, 5.97288852591006891113714963706, 6.96502749731653308834857677708, 7.42431133999694264228796352642, 8.370577673779143079998751821943
