L(s) = 1 | + 3-s − 3·5-s − 7-s + 9-s − 6·11-s − 13-s − 3·15-s − 8·17-s + 19-s − 21-s − 23-s + 4·25-s + 27-s − 5·29-s − 3·31-s − 6·33-s + 3·35-s + 12·37-s − 39-s + 10·41-s + 11·43-s − 3·45-s + 3·47-s + 49-s − 8·51-s − 53-s + 18·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s − 0.277·13-s − 0.774·15-s − 1.94·17-s + 0.229·19-s − 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s − 0.928·29-s − 0.538·31-s − 1.04·33-s + 0.507·35-s + 1.97·37-s − 0.160·39-s + 1.56·41-s + 1.67·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 1.12·51-s − 0.137·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8374495302\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8374495302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.217230044849672187426350618448, −7.57308680667792237098318336551, −7.35002271612004463854485746968, −6.24122901488136829757235859336, −5.31337573437497767589173093382, −4.33274626322043480479592219396, −3.95978471735366625224161879038, −2.75513972313535888128335811359, −2.35686577577771202149587602025, −0.46563938698910796465699560133,
0.46563938698910796465699560133, 2.35686577577771202149587602025, 2.75513972313535888128335811359, 3.95978471735366625224161879038, 4.33274626322043480479592219396, 5.31337573437497767589173093382, 6.24122901488136829757235859336, 7.35002271612004463854485746968, 7.57308680667792237098318336551, 8.217230044849672187426350618448