L(s) = 1 | − 3-s + 1.44·5-s − 3.44·7-s + 9-s + 5.18·11-s − 1.44·15-s − 0.753·17-s − 7.96·19-s + 3.44·21-s + 2.82·23-s − 2.91·25-s − 27-s − 3.91·29-s + 4.89·31-s − 5.18·33-s − 4.97·35-s + 6.24·37-s + 1.80·41-s + 7.09·43-s + 1.44·45-s − 10.5·47-s + 4.86·49-s + 0.753·51-s − 3.08·53-s + 7.49·55-s + 7.96·57-s − 1.87·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.646·5-s − 1.30·7-s + 0.333·9-s + 1.56·11-s − 0.373·15-s − 0.182·17-s − 1.82·19-s + 0.751·21-s + 0.589·23-s − 0.582·25-s − 0.192·27-s − 0.726·29-s + 0.880·31-s − 0.902·33-s − 0.841·35-s + 1.02·37-s + 0.281·41-s + 1.08·43-s + 0.215·45-s − 1.53·47-s + 0.695·49-s + 0.105·51-s − 0.424·53-s + 1.01·55-s + 1.05·57-s − 0.244·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435269757\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435269757\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 1.44T + 5T^{2} \) |
| 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 5.18T + 11T^{2} \) |
| 17 | \( 1 + 0.753T + 17T^{2} \) |
| 19 | \( 1 + 7.96T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.91T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 6.24T + 37T^{2} \) |
| 41 | \( 1 - 1.80T + 41T^{2} \) |
| 43 | \( 1 - 7.09T + 43T^{2} \) |
| 47 | \( 1 + 10.5T + 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 3.34T + 61T^{2} \) |
| 67 | \( 1 - 4.54T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 - 2.95T + 73T^{2} \) |
| 79 | \( 1 - 9.43T + 79T^{2} \) |
| 83 | \( 1 - 6.46T + 83T^{2} \) |
| 89 | \( 1 - 1.15T + 89T^{2} \) |
| 97 | \( 1 - 8.65T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.73241666878241980225825255398, −6.72592772762464610537725736689, −6.32662396791278320843069232898, −6.15902476914723246811740490215, −5.08462441173491190600183386260, −4.17548483887378871039336294142, −3.68512769217884323556025675854, −2.59554233738091422104566754205, −1.73043939602043243517770664676, −0.60855949483586979262469890311,
0.60855949483586979262469890311, 1.73043939602043243517770664676, 2.59554233738091422104566754205, 3.68512769217884323556025675854, 4.17548483887378871039336294142, 5.08462441173491190600183386260, 6.15902476914723246811740490215, 6.32662396791278320843069232898, 6.72592772762464610537725736689, 7.73241666878241980225825255398