| L(s) = 1 | + 3.40·3-s + 8.62·9-s − 3.62·11-s − 1.06·13-s + 5.75·17-s + 19.1·27-s + 9.62·29-s − 12.3·33-s − 3.62·39-s − 1.28·47-s + 19.6·51-s − 12·71-s + 13.4·73-s − 14.8·79-s + 39.4·81-s − 8.94·83-s + 32.8·87-s + 19.3·97-s − 31.2·99-s + 12.3·103-s + 20.8·109-s − 9.16·117-s + ⋯ |
| L(s) = 1 | + 1.96·3-s + 2.87·9-s − 1.09·11-s − 0.294·13-s + 1.39·17-s + 3.68·27-s + 1.78·29-s − 2.15·33-s − 0.580·39-s − 0.187·47-s + 2.74·51-s − 1.42·71-s + 1.57·73-s − 1.67·79-s + 4.38·81-s − 0.981·83-s + 3.51·87-s + 1.96·97-s − 3.14·99-s + 1.21·103-s + 1.99·109-s − 0.847·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.408116681\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.408116681\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 3.40T + 3T^{2} \) |
| 11 | \( 1 + 3.62T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 19.3T + 97T^{2} \) |
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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
−8.361098015759982694885617807468, −7.59474090593155258858580967686, −7.29616232850047869923809999356, −6.22707893434019680509820734961, −5.08844801340037788588471845931, −4.42866049153897732794754696209, −3.42053151735048252977608392448, −2.91686934021750974749645393946, −2.19893308165409539608203511014, −1.11891767210185429297561890902,
1.11891767210185429297561890902, 2.19893308165409539608203511014, 2.91686934021750974749645393946, 3.42053151735048252977608392448, 4.42866049153897732794754696209, 5.08844801340037788588471845931, 6.22707893434019680509820734961, 7.29616232850047869923809999356, 7.59474090593155258858580967686, 8.361098015759982694885617807468
