| L(s) = 1 | − 3-s − 5-s + 7-s + 9-s − 5·11-s − 13-s + 15-s − 7·17-s + 6·19-s − 21-s − 3·23-s + 25-s − 27-s + 2·29-s − 2·31-s + 5·33-s − 35-s + 7·37-s + 39-s + 9·41-s + 8·43-s − 45-s − 10·47-s − 6·49-s + 7·51-s + 5·53-s + 5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.258·15-s − 1.69·17-s + 1.37·19-s − 0.218·21-s − 0.625·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.359·31-s + 0.870·33-s − 0.169·35-s + 1.15·37-s + 0.160·39-s + 1.40·41-s + 1.21·43-s − 0.149·45-s − 1.45·47-s − 6/7·49-s + 0.980·51-s + 0.686·53-s + 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9647103016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9647103016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.593573881803824998352543452982, −7.75121566591838030416635069765, −7.39570009562781426019960665112, −6.38349246587353296444232484054, −5.57038387036564637510722191874, −4.82798919674137566892665441737, −4.23619202032551841583684345978, −2.98165958161286376777297354954, −2.10375385484976496829727724887, −0.59032891796402966674595042322,
0.59032891796402966674595042322, 2.10375385484976496829727724887, 2.98165958161286376777297354954, 4.23619202032551841583684345978, 4.82798919674137566892665441737, 5.57038387036564637510722191874, 6.38349246587353296444232484054, 7.39570009562781426019960665112, 7.75121566591838030416635069765, 8.593573881803824998352543452982
