| L(s) = 1 | − 3-s − 1.41·7-s + 9-s − 5.65·11-s + 5.65·13-s − 6.24·17-s + 0.242·19-s + 1.41·21-s − 23-s − 27-s + 3.17·29-s − 6.82·31-s + 5.65·33-s + 8.82·37-s − 5.65·39-s − 2·41-s + 2.58·43-s − 6.48·47-s − 5·49-s + 6.24·51-s − 7.41·53-s − 0.242·57-s − 3.17·59-s + 8.82·61-s − 1.41·63-s + 16.2·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.534·7-s + 0.333·9-s − 1.70·11-s + 1.56·13-s − 1.51·17-s + 0.0556·19-s + 0.308·21-s − 0.208·23-s − 0.192·27-s + 0.588·29-s − 1.22·31-s + 0.984·33-s + 1.45·37-s − 0.905·39-s − 0.312·41-s + 0.394·43-s − 0.945·47-s − 0.714·49-s + 0.874·51-s − 1.01·53-s − 0.0321·57-s − 0.412·59-s + 1.13·61-s − 0.178·63-s + 1.98·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8956055204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8956055204\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 19 | \( 1 - 0.242T + 19T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 8.82T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 7.41T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 16.2T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 1.07T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.000587832968689474573163436069, −7.19017502080190348089245233460, −6.38197314980088392883965349731, −5.99219196552607416625054790541, −5.15374485633334835982992231041, −4.46936146038804291465834022542, −3.58875737760548741977891932497, −2.74442105067997317864716789470, −1.80031216247949157214624840000, −0.48293955940911254918910959612,
0.48293955940911254918910959612, 1.80031216247949157214624840000, 2.74442105067997317864716789470, 3.58875737760548741977891932497, 4.46936146038804291465834022542, 5.15374485633334835982992231041, 5.99219196552607416625054790541, 6.38197314980088392883965349731, 7.19017502080190348089245233460, 8.000587832968689474573163436069
