L(s) = 1 | − 2.08·2-s + 2.36·4-s + 0.0546·5-s − 2.53·7-s − 0.756·8-s − 0.114·10-s + 11-s − 5.23·13-s + 5.29·14-s − 3.14·16-s + 1.86·17-s + 5.90·19-s + 0.129·20-s − 2.08·22-s + 3.02·23-s − 4.99·25-s + 10.9·26-s − 5.98·28-s + 5.01·29-s + 10.9·31-s + 8.08·32-s − 3.89·34-s − 0.138·35-s − 6.62·37-s − 12.3·38-s − 0.0413·40-s − 6.90·41-s + ⋯ |
L(s) = 1 | − 1.47·2-s + 1.18·4-s + 0.0244·5-s − 0.957·7-s − 0.267·8-s − 0.0360·10-s + 0.301·11-s − 1.45·13-s + 1.41·14-s − 0.786·16-s + 0.452·17-s + 1.35·19-s + 0.0288·20-s − 0.445·22-s + 0.630·23-s − 0.999·25-s + 2.14·26-s − 1.13·28-s + 0.930·29-s + 1.97·31-s + 1.42·32-s − 0.668·34-s − 0.0233·35-s − 1.08·37-s − 2.00·38-s − 0.00653·40-s − 1.07·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5950719568\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5950719568\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.0546T + 5T^{2} \) |
| 7 | \( 1 + 2.53T + 7T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 1.86T + 17T^{2} \) |
| 19 | \( 1 - 5.90T + 19T^{2} \) |
| 23 | \( 1 - 3.02T + 23T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 - 8.34T + 43T^{2} \) |
| 47 | \( 1 + 7.83T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 7.54T + 59T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 8.45T + 79T^{2} \) |
| 83 | \( 1 + 0.244T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−8.099364249116802281730717196136, −7.50829666062303500372623548365, −6.88093704976738662700518946921, −6.32809262009879256585833407710, −5.24064110156858865242667662787, −4.52894001160534294551146894347, −3.27498732903872761838538970259, −2.66487053293014265249669190505, −1.52751263870408392735377925478, −0.52302100148735806476472821151,
0.52302100148735806476472821151, 1.52751263870408392735377925478, 2.66487053293014265249669190505, 3.27498732903872761838538970259, 4.52894001160534294551146894347, 5.24064110156858865242667662787, 6.32809262009879256585833407710, 6.88093704976738662700518946921, 7.50829666062303500372623548365, 8.099364249116802281730717196136