| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s − 69·11-s + 12·12-s + 64·13-s − 14·14-s + 16·16-s + 114·17-s + 18·18-s + 56·19-s − 21·21-s − 138·22-s − 9·23-s + 24·24-s + 128·26-s + 27·27-s − 28·28-s − 33·29-s − 70·31-s + 32·32-s − 207·33-s + 228·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 1.89·11-s + 0.288·12-s + 1.36·13-s − 0.267·14-s + 1/4·16-s + 1.62·17-s + 0.235·18-s + 0.676·19-s − 0.218·21-s − 1.33·22-s − 0.0815·23-s + 0.204·24-s + 0.965·26-s + 0.192·27-s − 0.188·28-s − 0.211·29-s − 0.405·31-s + 0.176·32-s − 1.09·33-s + 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.177685260\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.177685260\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 69 T + p^{3} T^{2} \) |
| 13 | \( 1 - 64 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 - 56 T + p^{3} T^{2} \) |
| 23 | \( 1 + 9 T + p^{3} T^{2} \) |
| 29 | \( 1 + 33 T + p^{3} T^{2} \) |
| 31 | \( 1 + 70 T + p^{3} T^{2} \) |
| 37 | \( 1 + 53 T + p^{3} T^{2} \) |
| 41 | \( 1 - 504 T + p^{3} T^{2} \) |
| 43 | \( 1 + 137 T + p^{3} T^{2} \) |
| 47 | \( 1 - 600 T + p^{3} T^{2} \) |
| 53 | \( 1 - 570 T + p^{3} T^{2} \) |
| 59 | \( 1 - 48 T + p^{3} T^{2} \) |
| 61 | \( 1 - 524 T + p^{3} T^{2} \) |
| 67 | \( 1 - T + p^{3} T^{2} \) |
| 71 | \( 1 + 93 T + p^{3} T^{2} \) |
| 73 | \( 1 - 850 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1261 T + p^{3} T^{2} \) |
| 83 | \( 1 + 486 T + p^{3} T^{2} \) |
| 89 | \( 1 - 300 T + p^{3} T^{2} \) |
| 97 | \( 1 + 866 T + p^{3} T^{2} \) |
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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
−9.656591276631917962950630490767, −8.537880516711256315188789497130, −7.78266885832135275931540655729, −7.15624670172510432137438870152, −5.75600117983472126851722753379, −5.44837959688890247034704064906, −4.04781977087188144551500705607, −3.24085482599439135522417205530, −2.44244687751349458338599816995, −0.973129611253990780150763824960,
0.973129611253990780150763824960, 2.44244687751349458338599816995, 3.24085482599439135522417205530, 4.04781977087188144551500705607, 5.44837959688890247034704064906, 5.75600117983472126851722753379, 7.15624670172510432137438870152, 7.78266885832135275931540655729, 8.537880516711256315188789497130, 9.656591276631917962950630490767
