| L(s) = 1 | − 2.44·3-s + 5-s − 4.90·7-s + 2.97·9-s − 4.55·11-s + 3.11·13-s − 2.44·15-s − 4.54·17-s − 0.659·19-s + 11.9·21-s + 9.23·23-s + 25-s + 0.0713·27-s + 6.59·29-s − 8.71·31-s + 11.1·33-s − 4.90·35-s + 1.00·37-s − 7.62·39-s − 6.56·41-s + 4.24·43-s + 2.97·45-s + 5.91·47-s + 17.0·49-s + 11.1·51-s + 5.36·53-s − 4.55·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.447·5-s − 1.85·7-s + 0.990·9-s − 1.37·11-s + 0.865·13-s − 0.630·15-s − 1.10·17-s − 0.151·19-s + 2.61·21-s + 1.92·23-s + 0.200·25-s + 0.0137·27-s + 1.22·29-s − 1.56·31-s + 1.93·33-s − 0.829·35-s + 0.165·37-s − 1.22·39-s − 1.02·41-s + 0.647·43-s + 0.442·45-s + 0.863·47-s + 2.43·49-s + 1.55·51-s + 0.737·53-s − 0.614·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 151 | \( 1 - T \) |
| good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 + 4.55T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 + 4.54T + 17T^{2} \) |
| 19 | \( 1 + 0.659T + 19T^{2} \) |
| 23 | \( 1 - 9.23T + 23T^{2} \) |
| 29 | \( 1 - 6.59T + 29T^{2} \) |
| 31 | \( 1 + 8.71T + 31T^{2} \) |
| 37 | \( 1 - 1.00T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 - 4.24T + 43T^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 - 2.04T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 5.78T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 - 3.88T + 83T^{2} \) |
| 89 | \( 1 + 1.09T + 89T^{2} \) |
| 97 | \( 1 - 0.995T + 97T^{2} \) |
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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
−7.34214461066769593416011993203, −6.73574006314035104768236277012, −6.36205990909484382880494519869, −5.55207040783403873957535082391, −5.18960810924101566664058397113, −4.14693959782672172310147793955, −3.14407698905030335979847577830, −2.45195845603101473367589379587, −0.898756443363507894668429997423, 0,
0.898756443363507894668429997423, 2.45195845603101473367589379587, 3.14407698905030335979847577830, 4.14693959782672172310147793955, 5.18960810924101566664058397113, 5.55207040783403873957535082391, 6.36205990909484382880494519869, 6.73574006314035104768236277012, 7.34214461066769593416011993203
