| L(s) = 1 | − 2.90·3-s − 5-s − 4.42·7-s + 5.42·9-s + 2.62·11-s − 0.474·13-s + 2.90·15-s + 5.05·17-s + 19-s + 12.8·21-s − 1.37·23-s + 25-s − 7.05·27-s + 7.80·29-s + 1.24·31-s − 7.61·33-s + 4.42·35-s − 4.47·37-s + 1.37·39-s − 5.05·41-s − 12.0·43-s − 5.42·45-s − 4.42·47-s + 12.6·49-s − 14.6·51-s − 7.52·53-s − 2.62·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s − 0.447·5-s − 1.67·7-s + 1.80·9-s + 0.790·11-s − 0.131·13-s + 0.749·15-s + 1.22·17-s + 0.229·19-s + 2.80·21-s − 0.287·23-s + 0.200·25-s − 1.35·27-s + 1.44·29-s + 0.223·31-s − 1.32·33-s + 0.748·35-s − 0.735·37-s + 0.220·39-s − 0.788·41-s − 1.83·43-s − 0.809·45-s − 0.645·47-s + 1.80·49-s − 2.05·51-s − 1.03·53-s − 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5245028948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5245028948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.90T + 3T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 0.474T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 - 7.80T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 5.05T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 4.42T + 47T^{2} \) |
| 53 | \( 1 + 7.52T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 3.67T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 - 7.61T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.914456462506507280634595198364, −6.92249510959359585337383991409, −6.62477802172271175970176469951, −6.07201255088487225757331536331, −5.29641880870697287359443518363, −4.60147950978480512041118175424, −3.62695603368475438714263968206, −3.09063987814297246316168189228, −1.41980200327387179458143552122, −0.44370456335208626790131541677,
0.44370456335208626790131541677, 1.41980200327387179458143552122, 3.09063987814297246316168189228, 3.62695603368475438714263968206, 4.60147950978480512041118175424, 5.29641880870697287359443518363, 6.07201255088487225757331536331, 6.62477802172271175970176469951, 6.92249510959359585337383991409, 7.914456462506507280634595198364
