| L(s) = 1 | + 4.97·3-s + 5·5-s + 18.7·7-s − 2.27·9-s + 37.3·11-s + 70.5·13-s + 24.8·15-s + 72.1·17-s − 81.3·19-s + 93.0·21-s + 23·23-s + 25·25-s − 145.·27-s − 6.07·29-s + 276.·31-s + 185.·33-s + 93.5·35-s − 25.3·37-s + 350.·39-s + 169.·41-s − 141.·43-s − 11.3·45-s + 239.·47-s + 7.01·49-s + 358.·51-s − 151.·53-s + 186.·55-s + ⋯ |
| L(s) = 1 | + 0.956·3-s + 0.447·5-s + 1.01·7-s − 0.0842·9-s + 1.02·11-s + 1.50·13-s + 0.427·15-s + 1.02·17-s − 0.982·19-s + 0.966·21-s + 0.208·23-s + 0.200·25-s − 1.03·27-s − 0.0388·29-s + 1.59·31-s + 0.979·33-s + 0.451·35-s − 0.112·37-s + 1.44·39-s + 0.645·41-s − 0.500·43-s − 0.0376·45-s + 0.742·47-s + 0.0204·49-s + 0.984·51-s − 0.393·53-s + 0.457·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.789909861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.789909861\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 4.97T + 27T^{2} \) |
| 7 | \( 1 - 18.7T + 343T^{2} \) |
| 11 | \( 1 - 37.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.3T + 6.85e3T^{2} \) |
| 29 | \( 1 + 6.07T + 2.43e4T^{2} \) |
| 31 | \( 1 - 276.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 25.3T + 5.06e4T^{2} \) |
| 41 | \( 1 - 169.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 239.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 151.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 800.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 164.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 248.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 729.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 907.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 599.T + 9.12e5T^{2} \) |
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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
−8.678588706178038507883966242830, −8.368668479530918208130792405029, −7.55635848028793907838117599050, −6.38600502304364862274514435820, −5.83511134285886323507824774326, −4.64797710253682654197896215874, −3.80609299423795150932461897951, −2.93848302721001219361121066054, −1.81213387748317740276048556267, −1.09094012821478903857989931933,
1.09094012821478903857989931933, 1.81213387748317740276048556267, 2.93848302721001219361121066054, 3.80609299423795150932461897951, 4.64797710253682654197896215874, 5.83511134285886323507824774326, 6.38600502304364862274514435820, 7.55635848028793907838117599050, 8.368668479530918208130792405029, 8.678588706178038507883966242830
