| L(s) = 1 | + 3-s + 4.64·7-s + 9-s − 6.12·11-s − 3.47·13-s + 6.44·17-s + 4.64·21-s − 6.44·23-s + 27-s + 8.96·29-s + 31-s − 6.12·33-s + 2.64·37-s − 3.47·39-s + 8.12·41-s − 6.12·43-s − 0.853·47-s + 14.6·49-s + 6.44·51-s + 1.14·53-s − 10.5·59-s + 1.36·61-s + 4.64·63-s − 6.11·67-s − 6.44·69-s + 3.79·71-s + 5.60·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.75·7-s + 0.333·9-s − 1.84·11-s − 0.964·13-s + 1.56·17-s + 1.01·21-s − 1.34·23-s + 0.192·27-s + 1.66·29-s + 0.179·31-s − 1.06·33-s + 0.435·37-s − 0.556·39-s + 1.26·41-s − 0.934·43-s − 0.124·47-s + 2.08·49-s + 0.902·51-s + 0.157·53-s − 1.37·59-s + 0.174·61-s + 0.585·63-s − 0.747·67-s − 0.776·69-s + 0.450·71-s + 0.655·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.959342430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.959342430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 4.64T + 7T^{2} \) |
| 11 | \( 1 + 6.12T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 6.44T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 8.96T + 29T^{2} \) |
| 37 | \( 1 - 2.64T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 + 0.853T + 47T^{2} \) |
| 53 | \( 1 - 1.14T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 - 3.79T + 71T^{2} \) |
| 73 | \( 1 - 5.60T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 9.79T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.73216747410961357470247476688, −7.60780204597131383693783599858, −6.29598032191077637935597187038, −5.41415949092691660272987935302, −4.92510313577196996197555024937, −4.43152464116169166681566033788, −3.29418249909684673762909830770, −2.49917137286664821357728910185, −1.93614072579941837565791835921, −0.805112380110933515728906409191,
0.805112380110933515728906409191, 1.93614072579941837565791835921, 2.49917137286664821357728910185, 3.29418249909684673762909830770, 4.43152464116169166681566033788, 4.92510313577196996197555024937, 5.41415949092691660272987935302, 6.29598032191077637935597187038, 7.60780204597131383693783599858, 7.73216747410961357470247476688
