| L(s) = 1 | − 5·5-s + 32·7-s − 64·11-s − 6·13-s − 38·17-s − 116·19-s + 120·23-s + 25·25-s + 122·29-s + 164·31-s − 160·35-s + 146·37-s + 238·41-s − 148·43-s + 184·47-s + 681·49-s − 470·53-s + 320·55-s + 216·59-s + 806·61-s + 30·65-s − 732·67-s − 264·71-s − 638·73-s − 2.04e3·77-s + 596·79-s + 884·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.72·7-s − 1.75·11-s − 0.128·13-s − 0.542·17-s − 1.40·19-s + 1.08·23-s + 1/5·25-s + 0.781·29-s + 0.950·31-s − 0.772·35-s + 0.648·37-s + 0.906·41-s − 0.524·43-s + 0.571·47-s + 1.98·49-s − 1.21·53-s + 0.784·55-s + 0.476·59-s + 1.69·61-s + 0.0572·65-s − 1.33·67-s − 0.441·71-s − 1.02·73-s − 3.03·77-s + 0.848·79-s + 1.16·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.939667390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939667390\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 + 38 T + p^{3} T^{2} \) |
| 19 | \( 1 + 116 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 164 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 238 T + p^{3} T^{2} \) |
| 43 | \( 1 + 148 T + p^{3} T^{2} \) |
| 47 | \( 1 - 184 T + p^{3} T^{2} \) |
| 53 | \( 1 + 470 T + p^{3} T^{2} \) |
| 59 | \( 1 - 216 T + p^{3} T^{2} \) |
| 61 | \( 1 - 806 T + p^{3} T^{2} \) |
| 67 | \( 1 + 732 T + p^{3} T^{2} \) |
| 71 | \( 1 + 264 T + p^{3} T^{2} \) |
| 73 | \( 1 + 638 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 - 884 T + p^{3} T^{2} \) |
| 89 | \( 1 + 930 T + p^{3} T^{2} \) |
| 97 | \( 1 - 322 T + p^{3} T^{2} \) |
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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
−8.870017360566454134041475571094, −8.222865535898623116303426649794, −7.79898643419207128714246271461, −6.88299930433256774026635223295, −5.69765821472405700922559400059, −4.74018066909448217879659390512, −4.45795335143521183959730068488, −2.84718238724257255418807461441, −2.04211362598052834391328688738, −0.67913846168074242709035281313,
0.67913846168074242709035281313, 2.04211362598052834391328688738, 2.84718238724257255418807461441, 4.45795335143521183959730068488, 4.74018066909448217879659390512, 5.69765821472405700922559400059, 6.88299930433256774026635223295, 7.79898643419207128714246271461, 8.222865535898623116303426649794, 8.870017360566454134041475571094
