| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 5·11-s + 12-s + 4·13-s − 16-s − 4·17-s − 18-s − 19-s − 5·22-s − 9·23-s − 3·24-s − 4·26-s − 27-s + 7·29-s + 3·31-s − 5·32-s − 5·33-s + 4·34-s − 36-s − 10·37-s + 38-s − 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.229·19-s − 1.06·22-s − 1.87·23-s − 0.612·24-s − 0.784·26-s − 0.192·27-s + 1.29·29-s + 0.538·31-s − 0.883·32-s − 0.870·33-s + 0.685·34-s − 1/6·36-s − 1.64·37-s + 0.162·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8415530260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8415530260\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.561441370849687579716007625047, −8.559969876361858219788163822976, −8.399919956373444667684985492191, −6.98515498693075208426360294141, −6.44029770462908743404015128774, −5.47270146644958024862708785711, −4.24850220460461882940935059602, −3.87301531439376933769109060020, −1.89916774879802682118008269843, −0.794038070961207158407211508808,
0.794038070961207158407211508808, 1.89916774879802682118008269843, 3.87301531439376933769109060020, 4.24850220460461882940935059602, 5.47270146644958024862708785711, 6.44029770462908743404015128774, 6.98515498693075208426360294141, 8.399919956373444667684985492191, 8.559969876361858219788163822976, 9.561441370849687579716007625047
