| L(s) = 1 | − 3·3-s + 18.8·5-s − 17.9·7-s + 9·9-s + 44.5·11-s + 13·13-s − 56.6·15-s + 34·17-s − 152.·19-s + 53.7·21-s + 107.·23-s + 231.·25-s − 27·27-s + 149.·29-s + 0.162·31-s − 133.·33-s − 338.·35-s − 256.·37-s − 39·39-s + 414.·41-s + 471.·43-s + 169.·45-s + 632.·47-s − 21.6·49-s − 102·51-s − 236.·53-s + 841.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.68·5-s − 0.967·7-s + 0.333·9-s + 1.22·11-s + 0.277·13-s − 0.975·15-s + 0.485·17-s − 1.84·19-s + 0.558·21-s + 0.972·23-s + 1.85·25-s − 0.192·27-s + 0.958·29-s + 0.000943·31-s − 0.705·33-s − 1.63·35-s − 1.13·37-s − 0.160·39-s + 1.57·41-s + 1.67·43-s + 0.562·45-s + 1.96·47-s − 0.0630·49-s − 0.280·51-s − 0.612·53-s + 2.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.025862988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.025862988\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 18.8T + 125T^{2} \) |
| 7 | \( 1 + 17.9T + 343T^{2} \) |
| 11 | \( 1 - 44.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 34T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 0.162T + 2.97e4T^{2} \) |
| 37 | \( 1 + 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 414.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 632.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 236.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 108.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 888.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 637.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 362.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 723.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 964.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 431.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 117.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.15e3T + 9.12e5T^{2} \) |
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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
−11.01788255615281080044484688719, −10.29021215663515932674085102953, −9.434161759233823153514538101846, −8.795359260173360243040948108278, −6.85588928254275163404796722296, −6.31925337851100174766700954760, −5.55812580269464539366589043322, −4.10833045461729011644367023953, −2.49479002275101306809856162620, −1.08246744018822829343940846184,
1.08246744018822829343940846184, 2.49479002275101306809856162620, 4.10833045461729011644367023953, 5.55812580269464539366589043322, 6.31925337851100174766700954760, 6.85588928254275163404796722296, 8.795359260173360243040948108278, 9.434161759233823153514538101846, 10.29021215663515932674085102953, 11.01788255615281080044484688719
