| L(s) = 1 | + 3-s + 0.284·5-s − 3.90·7-s + 9-s + 5.93·11-s + 5.62·13-s + 0.284·15-s − 0.211·17-s − 1.66·19-s − 3.90·21-s − 2.16·23-s − 4.91·25-s + 27-s + 4.18·29-s − 2.41·31-s + 5.93·33-s − 1.10·35-s + 0.0651·37-s + 5.62·39-s + 6.23·41-s + 2.49·43-s + 0.284·45-s + 5.97·47-s + 8.26·49-s − 0.211·51-s + 4.43·53-s + 1.68·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.127·5-s − 1.47·7-s + 0.333·9-s + 1.79·11-s + 1.55·13-s + 0.0733·15-s − 0.0513·17-s − 0.381·19-s − 0.852·21-s − 0.451·23-s − 0.983·25-s + 0.192·27-s + 0.777·29-s − 0.433·31-s + 1.03·33-s − 0.187·35-s + 0.0107·37-s + 0.900·39-s + 0.973·41-s + 0.379·43-s + 0.0423·45-s + 0.871·47-s + 1.18·49-s − 0.0296·51-s + 0.609·53-s + 0.227·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.431945108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431945108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.284T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 - 5.93T + 11T^{2} \) |
| 13 | \( 1 - 5.62T + 13T^{2} \) |
| 17 | \( 1 + 0.211T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 + 2.41T + 31T^{2} \) |
| 37 | \( 1 - 0.0651T + 37T^{2} \) |
| 41 | \( 1 - 6.23T + 41T^{2} \) |
| 43 | \( 1 - 2.49T + 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 + 5.08T + 67T^{2} \) |
| 71 | \( 1 - 6.06T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 + 5.50T + 83T^{2} \) |
| 89 | \( 1 - 0.660T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−8.748039866212112340414516422957, −7.73216118529967578273206759066, −6.82282964907565005935130023796, −6.24365545735844081707740852810, −5.87825836746920061069968677542, −4.24809214497299685837603974734, −3.80848124644932924212980020187, −3.15044959160528921052670387400, −1.98356456576968556064793672945, −0.900512101904615953438219672147,
0.900512101904615953438219672147, 1.98356456576968556064793672945, 3.15044959160528921052670387400, 3.80848124644932924212980020187, 4.24809214497299685837603974734, 5.87825836746920061069968677542, 6.24365545735844081707740852810, 6.82282964907565005935130023796, 7.73216118529967578273206759066, 8.748039866212112340414516422957
