| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s − 2.44·7-s − 2.82i·8-s + 3.46i·11-s + (2.99 + 1.73i)14-s + (−2.00 + 3.46i)16-s − 4.89·17-s − 3.46i·19-s + (2.44 − 4.24i)22-s + 2.44·23-s + (−2.44 − 4.24i)28-s + 4·31-s + (4.89 − 2.82i)32-s + (5.99 + 3.46i)34-s + ⋯ |
| L(s) = 1 | + (−0.866 − 0.499i)2-s + (0.499 + 0.866i)4-s − 0.925·7-s − 0.999i·8-s + 1.04i·11-s + (0.801 + 0.462i)14-s + (−0.500 + 0.866i)16-s − 1.18·17-s − 0.794i·19-s + (0.522 − 0.904i)22-s + 0.510·23-s + (−0.462 − 0.801i)28-s + 0.718·31-s + (0.866 − 0.499i)32-s + (1.02 + 0.594i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6766387980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6766387980\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 2.44T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 - 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4.89T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 9.89iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89T + 97T^{2} \) |
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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
−9.226251592886954824248793779419, −8.555930012902683377214918062286, −7.42800222419504607409704551065, −6.93749542442149101339125883601, −6.19009418774069882560641619140, −4.78092956179463461541593903246, −3.89304098357491775614795993044, −2.80928038913743928673371212771, −2.00885248149841967506496376856, −0.41147841483280687946485728547,
0.959450529265815161385202727685, 2.43034099500568851557385378476, 3.42364162315385665145149840777, 4.71718787653922783839810085779, 5.80566145363091750754294369468, 6.39723393862334543260024385676, 7.01853775513162350390606492938, 8.097312940011261414792853551354, 8.627592815640512230679452424112, 9.401186723544113542833334599493
