L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 6·11-s + 12-s − 4·13-s + 16-s − 3·17-s + 18-s − 4·19-s + 6·22-s + 3·23-s + 24-s − 4·26-s + 27-s − 6·29-s + 5·31-s + 32-s + 6·33-s − 3·34-s + 36-s + 8·37-s − 4·38-s − 4·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.80·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.917·19-s + 1.27·22-s + 0.625·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 1.11·29-s + 0.898·31-s + 0.176·32-s + 1.04·33-s − 0.514·34-s + 1/6·36-s + 1.31·37-s − 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.477720744\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.477720744\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.72942965885478190670686461883, −7.08612274658929900256725680230, −6.57299487250174225949745822834, −5.86579865569663357298211589102, −4.89429112300728681632615412802, −4.13824418679496906880697490543, −3.82918568191424993418746399940, −2.63033734901572427314822965104, −2.12581274247356368224220798439, −0.955207885496938579671476750691,
0.955207885496938579671476750691, 2.12581274247356368224220798439, 2.63033734901572427314822965104, 3.82918568191424993418746399940, 4.13824418679496906880697490543, 4.89429112300728681632615412802, 5.86579865569663357298211589102, 6.57299487250174225949745822834, 7.08612274658929900256725680230, 7.72942965885478190670686461883