L(s) = 1 | − 0.486·3-s − 1.80·7-s − 2.76·9-s − 2.90·11-s + 2.25·13-s − 2.14·17-s + 0.339·19-s + 0.877·21-s + 23-s + 2.80·27-s − 5.60·29-s + 5.92·31-s + 1.41·33-s + 8.98·37-s − 1.09·39-s + 1.89·41-s + 9.47·43-s + 7.83·47-s − 3.74·49-s + 1.04·51-s − 6.47·53-s − 0.165·57-s + 5.17·59-s − 9.12·61-s + 4.98·63-s + 9.25·67-s − 0.486·69-s + ⋯ |
L(s) = 1 | − 0.280·3-s − 0.682·7-s − 0.921·9-s − 0.876·11-s + 0.624·13-s − 0.520·17-s + 0.0779·19-s + 0.191·21-s + 0.208·23-s + 0.539·27-s − 1.04·29-s + 1.06·31-s + 0.246·33-s + 1.47·37-s − 0.175·39-s + 0.295·41-s + 1.44·43-s + 1.14·47-s − 0.534·49-s + 0.146·51-s − 0.889·53-s − 0.0218·57-s + 0.674·59-s − 1.16·61-s + 0.628·63-s + 1.13·67-s − 0.0585·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.074906221\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074906221\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 0.486T + 3T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 19 | \( 1 - 0.339T + 19T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 - 8.98T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 7.83T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 9.12T + 61T^{2} \) |
| 67 | \( 1 - 9.25T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 7.94T + 79T^{2} \) |
| 83 | \( 1 + 5.37T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.43T + 97T^{2} \) |
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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
−9.086011716501989311148646539026, −8.200581938274389029970461831825, −7.55965781110712084181711752565, −6.45935106412332657950633888018, −5.96715882656471800385461674845, −5.16653274452085357263550380908, −4.15400298132527953395392568166, −3.11021124616191614478091097437, −2.36003935288296194660693204380, −0.65301526476596052360718351039,
0.65301526476596052360718351039, 2.36003935288296194660693204380, 3.11021124616191614478091097437, 4.15400298132527953395392568166, 5.16653274452085357263550380908, 5.96715882656471800385461674845, 6.45935106412332657950633888018, 7.55965781110712084181711752565, 8.200581938274389029970461831825, 9.086011716501989311148646539026