| L(s) = 1 | + 3·3-s − 9.46·7-s + 9·9-s − 55.0·11-s − 68.5·13-s − 22.1·17-s − 15.1·19-s − 28.3·21-s + 67.0·23-s + 27·27-s + 102.·29-s − 89.3·31-s − 165.·33-s + 96.7·37-s − 205.·39-s − 266.·41-s + 143.·43-s − 241.·47-s − 253.·49-s − 66.4·51-s − 554.·53-s − 45.4·57-s + 579.·59-s + 543.·61-s − 85.1·63-s + 784.·67-s + 201.·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.510·7-s + 0.333·9-s − 1.50·11-s − 1.46·13-s − 0.316·17-s − 0.183·19-s − 0.295·21-s + 0.608·23-s + 0.192·27-s + 0.656·29-s − 0.517·31-s − 0.871·33-s + 0.430·37-s − 0.844·39-s − 1.01·41-s + 0.509·43-s − 0.749·47-s − 0.738·49-s − 0.182·51-s − 1.43·53-s − 0.105·57-s + 1.27·59-s + 1.14·61-s − 0.170·63-s + 1.42·67-s + 0.351·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.442327332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.442327332\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 9.46T + 343T^{2} \) |
| 11 | \( 1 + 55.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 67.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 102.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 89.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 96.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 143.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 241.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 554.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 579.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 543.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 784.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 712.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 795.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 485.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 3.75T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.26e3T + 9.12e5T^{2} \) |
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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
−8.481711193873491306604835251557, −7.909728807845785937841381405761, −7.15459216983265502161633138191, −6.47026633118985050763742672557, −5.21482789915014175579989154153, −4.81550024900305198060479079540, −3.56675371915153537845087041199, −2.71770438914604448842693103784, −2.10820769236333501015039782289, −0.48397815413661429340336453229,
0.48397815413661429340336453229, 2.10820769236333501015039782289, 2.71770438914604448842693103784, 3.56675371915153537845087041199, 4.81550024900305198060479079540, 5.21482789915014175579989154153, 6.47026633118985050763742672557, 7.15459216983265502161633138191, 7.909728807845785937841381405761, 8.481711193873491306604835251557
