| L(s) = 1 | + 2·2-s + 4·4-s + 16.2·5-s − 14.9·7-s + 8·8-s + 32.4·10-s − 40.5·13-s − 29.8·14-s + 16·16-s + 20.6·17-s − 127.·19-s + 64.9·20-s + 66.9·23-s + 139.·25-s − 81.1·26-s − 59.6·28-s − 81.4·29-s − 175.·31-s + 32·32-s + 41.3·34-s − 242.·35-s − 274.·37-s − 255.·38-s + 129.·40-s − 376.·41-s + 97.4·43-s + 133.·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.45·5-s − 0.805·7-s + 0.353·8-s + 1.02·10-s − 0.865·13-s − 0.569·14-s + 0.250·16-s + 0.294·17-s − 1.54·19-s + 0.726·20-s + 0.606·23-s + 1.11·25-s − 0.612·26-s − 0.402·28-s − 0.521·29-s − 1.01·31-s + 0.176·32-s + 0.208·34-s − 1.17·35-s − 1.21·37-s − 1.09·38-s + 0.513·40-s − 1.43·41-s + 0.345·43-s + 0.428·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 16.2T + 125T^{2} \) |
| 7 | \( 1 + 14.9T + 343T^{2} \) |
| 13 | \( 1 + 40.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 274.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 376.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 28.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 479.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 815.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 781.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 809.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 798.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 842.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 271.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−8.462016018380642974751492292105, −7.06957675915272490182093809220, −6.76352359922701001897936316792, −5.75496628986292702423558544108, −5.37241001003209315176309277517, −4.31471846646229394760900770458, −3.26291051580129182133677355617, −2.37553582244623737216676283516, −1.64870113725426953666378378412, 0,
1.64870113725426953666378378412, 2.37553582244623737216676283516, 3.26291051580129182133677355617, 4.31471846646229394760900770458, 5.37241001003209315176309277517, 5.75496628986292702423558544108, 6.76352359922701001897936316792, 7.06957675915272490182093809220, 8.462016018380642974751492292105
