| L(s) = 1 | + 3-s − 5-s − 3.24·7-s + 9-s − 1.35·11-s + 1.59·13-s − 15-s − 1.89·17-s + 6.84·19-s − 3.24·21-s − 4.02·23-s + 25-s + 27-s + 2.30·29-s + 31-s − 1.35·33-s + 3.24·35-s − 8.08·37-s + 1.59·39-s + 3.08·41-s + 7.92·43-s − 45-s + 5.38·47-s + 3.54·49-s − 1.89·51-s − 11.3·53-s + 1.35·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.22·7-s + 0.333·9-s − 0.408·11-s + 0.442·13-s − 0.258·15-s − 0.460·17-s + 1.56·19-s − 0.708·21-s − 0.839·23-s + 0.200·25-s + 0.192·27-s + 0.427·29-s + 0.179·31-s − 0.236·33-s + 0.548·35-s − 1.32·37-s + 0.255·39-s + 0.481·41-s + 1.20·43-s − 0.149·45-s + 0.785·47-s + 0.505·49-s − 0.265·51-s − 1.55·53-s + 0.182·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666235696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666235696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 3.24T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 - 1.59T + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 6.84T + 19T^{2} \) |
| 23 | \( 1 + 4.02T + 23T^{2} \) |
| 29 | \( 1 - 2.30T + 29T^{2} \) |
| 37 | \( 1 + 8.08T + 37T^{2} \) |
| 41 | \( 1 - 3.08T + 41T^{2} \) |
| 43 | \( 1 - 7.92T + 43T^{2} \) |
| 47 | \( 1 - 5.38T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 - 5.78T + 59T^{2} \) |
| 61 | \( 1 - 0.514T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 5.43T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 - 6.83T + 89T^{2} \) |
| 97 | \( 1 + 5.57T + 97T^{2} \) |
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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
−8.496627733569555947093207492667, −7.79220919149956146180311109372, −7.12872863350582141642557721207, −6.39137184309317577852354884356, −5.59115139604670223260283075187, −4.59803215270567025949538803733, −3.61293583086676560652927446779, −3.17996593809024746484793246303, −2.17711445951069111187383185610, −0.71479527836383060864649423215,
0.71479527836383060864649423215, 2.17711445951069111187383185610, 3.17996593809024746484793246303, 3.61293583086676560652927446779, 4.59803215270567025949538803733, 5.59115139604670223260283075187, 6.39137184309317577852354884356, 7.12872863350582141642557721207, 7.79220919149956146180311109372, 8.496627733569555947093207492667
