| L(s) = 1 | + 2-s + 4-s − 3.17·5-s − 4.10·7-s + 8-s − 3.17·10-s − 5.68·11-s − 6.22·13-s − 4.10·14-s + 16-s − 5.79·17-s + 2.64·19-s − 3.17·20-s − 5.68·22-s + 6.44·23-s + 5.07·25-s − 6.22·26-s − 4.10·28-s − 3.80·29-s + 2.24·31-s + 32-s − 5.79·34-s + 13.0·35-s + 7.14·37-s + 2.64·38-s − 3.17·40-s − 6.53·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.41·5-s − 1.55·7-s + 0.353·8-s − 1.00·10-s − 1.71·11-s − 1.72·13-s − 1.09·14-s + 0.250·16-s − 1.40·17-s + 0.607·19-s − 0.709·20-s − 1.21·22-s + 1.34·23-s + 1.01·25-s − 1.22·26-s − 0.776·28-s − 0.706·29-s + 0.403·31-s + 0.176·32-s − 0.994·34-s + 2.20·35-s + 1.17·37-s + 0.429·38-s − 0.501·40-s − 1.01·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4767365161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4767365161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + 6.22T + 13T^{2} \) |
| 17 | \( 1 + 5.79T + 17T^{2} \) |
| 19 | \( 1 - 2.64T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 3.49T + 47T^{2} \) |
| 53 | \( 1 + 7.70T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 6.40T + 61T^{2} \) |
| 67 | \( 1 + 4.28T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 6.85T + 73T^{2} \) |
| 79 | \( 1 - 8.62T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 0.930T + 89T^{2} \) |
| 97 | \( 1 - 6.12T + 97T^{2} \) |
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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
−7.890332009391394036026504882879, −7.49486187148367735732764179783, −6.92152114913212296774298917563, −6.14906404819283745577787603425, −5.00201843978366650152707984262, −4.72894618317972099298728037649, −3.67401759154476916894254039589, −2.91782851994414119209365885329, −2.49602315598841830090023976631, −0.30963607385756339419788256684,
0.30963607385756339419788256684, 2.49602315598841830090023976631, 2.91782851994414119209365885329, 3.67401759154476916894254039589, 4.72894618317972099298728037649, 5.00201843978366650152707984262, 6.14906404819283745577787603425, 6.92152114913212296774298917563, 7.49486187148367735732764179783, 7.890332009391394036026504882879
