| L(s) = 1 | − 4·2-s − 15.8·3-s + 16·4-s + 21·5-s + 63.2·6-s − 64·8-s + 6.75·9-s − 84·10-s − 625.·11-s − 252.·12-s + 206.·13-s − 331.·15-s + 256·16-s − 1.06e3·17-s − 27.0·18-s + 1.88e3·19-s + 336·20-s + 2.50e3·22-s + 3.71e3·23-s + 1.01e3·24-s − 2.68e3·25-s − 827.·26-s + 3.73e3·27-s − 123.·29-s + 1.32e3·30-s + 9.10e3·31-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.01·3-s + 0.5·4-s + 0.375·5-s + 0.716·6-s − 0.353·8-s + 0.0277·9-s − 0.265·10-s − 1.55·11-s − 0.506·12-s + 0.339·13-s − 0.380·15-s + 0.250·16-s − 0.890·17-s − 0.0196·18-s + 1.19·19-s + 0.187·20-s + 1.10·22-s + 1.46·23-s + 0.358·24-s − 0.858·25-s − 0.240·26-s + 0.985·27-s − 0.0273·29-s + 0.269·30-s + 1.70·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7445610215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7445610215\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 15.8T + 243T^{2} \) |
| 5 | \( 1 - 21T + 3.12e3T^{2} \) |
| 11 | \( 1 + 625.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 206.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.88e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 123.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.10e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.02e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.72e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.87e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.53e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.09e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.15e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.97e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.59e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−12.81709534689572315232736598469, −11.56098116642759131289371596203, −10.83913495993302690137834504632, −9.898934573701429047103416194585, −8.589232309011873197609852457946, −7.31375942334279394980130482519, −6.02481892948640697300908545664, −5.04945301378655417418616814790, −2.67208281501642659223431370480, −0.71401889630299768595850059387,
0.71401889630299768595850059387, 2.67208281501642659223431370480, 5.04945301378655417418616814790, 6.02481892948640697300908545664, 7.31375942334279394980130482519, 8.589232309011873197609852457946, 9.898934573701429047103416194585, 10.83913495993302690137834504632, 11.56098116642759131289371596203, 12.81709534689572315232736598469
