| L(s) = 1 | + 5·5-s + 6.80·7-s + 39.2·11-s + 78.4·13-s + 95.2·17-s + 133.·19-s + 66.8·23-s + 25·25-s − 99.6·29-s − 322.·31-s + 34.0·35-s + 108.·37-s − 278.·41-s − 381.·43-s + 211.·47-s − 296.·49-s + 411.·53-s + 196.·55-s + 447.·59-s + 158.·61-s + 392.·65-s − 455.·67-s + 630.·71-s + 58.8·73-s + 266.·77-s − 1.25e3·79-s − 229.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.367·7-s + 1.07·11-s + 1.67·13-s + 1.35·17-s + 1.60·19-s + 0.605·23-s + 0.200·25-s − 0.638·29-s − 1.86·31-s + 0.164·35-s + 0.483·37-s − 1.05·41-s − 1.35·43-s + 0.656·47-s − 0.864·49-s + 1.06·53-s + 0.480·55-s + 0.986·59-s + 0.333·61-s + 0.748·65-s − 0.830·67-s + 1.05·71-s + 0.0943·73-s + 0.395·77-s − 1.79·79-s − 0.303·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.398221969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398221969\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 6.80T + 343T^{2} \) |
| 11 | \( 1 - 39.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 99.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 322.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 108.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 381.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 211.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 411.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 447.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 158.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 455.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 630.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 58.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 229.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.17e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−9.177049841978959909785187118649, −8.469716995451169194895440710885, −7.52082363953580185764170426539, −6.72591911449694877846000319197, −5.73475544578824587362861421491, −5.23636867022084971006085676178, −3.79950407306486742001604928767, −3.29529289050615422368511432981, −1.63637426190762657288974081332, −1.04735327972372327641819157958,
1.04735327972372327641819157958, 1.63637426190762657288974081332, 3.29529289050615422368511432981, 3.79950407306486742001604928767, 5.23636867022084971006085676178, 5.73475544578824587362861421491, 6.72591911449694877846000319197, 7.52082363953580185764170426539, 8.469716995451169194895440710885, 9.177049841978959909785187118649
