| L(s) = 1 | − 6.84·3-s + 5·5-s + 28.6·7-s + 19.9·9-s − 26.1·11-s + 74.7·13-s − 34.2·15-s + 121.·17-s + 6.41·19-s − 195.·21-s − 23·23-s + 25·25-s + 48.5·27-s + 281.·29-s + 36.7·31-s + 178.·33-s + 143.·35-s + 287.·37-s − 511.·39-s − 435.·41-s + 54.2·43-s + 99.5·45-s + 448.·47-s + 475.·49-s − 831.·51-s + 488.·53-s − 130.·55-s + ⋯ |
| L(s) = 1 | − 1.31·3-s + 0.447·5-s + 1.54·7-s + 0.737·9-s − 0.716·11-s + 1.59·13-s − 0.589·15-s + 1.73·17-s + 0.0775·19-s − 2.03·21-s − 0.208·23-s + 0.200·25-s + 0.346·27-s + 1.80·29-s + 0.212·31-s + 0.943·33-s + 0.690·35-s + 1.27·37-s − 2.10·39-s − 1.65·41-s + 0.192·43-s + 0.329·45-s + 1.39·47-s + 1.38·49-s − 2.28·51-s + 1.26·53-s − 0.320·55-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
| \(L(2)\) |
\(\approx\) |
\(2.269807182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269807182\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 6.84T + 27T^{2} \) |
| 7 | \( 1 - 28.6T + 343T^{2} \) |
| 11 | \( 1 + 26.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.41T + 6.85e3T^{2} \) |
| 29 | \( 1 - 281.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 36.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 287.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 54.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 448.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 488.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 84.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 293.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 399.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 870.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 64.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 262.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 995.T + 9.12e5T^{2} \) |
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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
−8.642560714363047724488489495400, −8.180944833265139456087839675447, −7.28842797013175320591611970105, −6.17552084169276556576926297340, −5.65840917769170590971574409134, −5.05917709814010199735759596402, −4.22011412087632645887610803642, −2.86391743179380916881353256562, −1.42224495908066620719252778918, −0.881834314545339318489390208176,
0.881834314545339318489390208176, 1.42224495908066620719252778918, 2.86391743179380916881353256562, 4.22011412087632645887610803642, 5.05917709814010199735759596402, 5.65840917769170590971574409134, 6.17552084169276556576926297340, 7.28842797013175320591611970105, 8.180944833265139456087839675447, 8.642560714363047724488489495400
