L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 13-s + 15-s + 6·19-s + 2·21-s + 2·23-s + 25-s − 27-s − 4·29-s − 8·31-s + 2·35-s − 10·37-s + 39-s − 2·41-s + 12·43-s − 45-s + 4·47-s − 3·49-s + 6·53-s − 6·57-s − 8·59-s − 2·61-s − 2·63-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 1.37·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s + 0.160·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.794·57-s − 1.04·59-s − 0.256·61-s − 0.251·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9477548306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9477548306\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.84159260764064842968865159418, −7.19773607781556682912964485518, −6.83613644301337970943682406526, −5.69669129713303018125356379802, −5.42491641941761392565163766594, −4.40954908194062381260735833941, −3.60513748978311367021979424100, −2.95584723672102699188506946539, −1.71348486018265623815718068995, −0.52144196617683708403193604902,
0.52144196617683708403193604902, 1.71348486018265623815718068995, 2.95584723672102699188506946539, 3.60513748978311367021979424100, 4.40954908194062381260735833941, 5.42491641941761392565163766594, 5.69669129713303018125356379802, 6.83613644301337970943682406526, 7.19773607781556682912964485518, 7.84159260764064842968865159418