L(s) = 1 | + 3-s − 5-s + 3.61·7-s + 9-s − 3.95·11-s + 5.10·13-s − 15-s + 4.16·17-s + 2.79·19-s + 3.61·21-s + 3.13·23-s + 25-s + 27-s − 1.00·29-s + 2.03·31-s − 3.95·33-s − 3.61·35-s − 6.39·37-s + 5.10·39-s − 0.0315·41-s + 1.97·43-s − 45-s + 3.94·47-s + 6.03·49-s + 4.16·51-s − 2.86·53-s + 3.95·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.36·7-s + 0.333·9-s − 1.19·11-s + 1.41·13-s − 0.258·15-s + 1.01·17-s + 0.640·19-s + 0.787·21-s + 0.653·23-s + 0.200·25-s + 0.192·27-s − 0.186·29-s + 0.364·31-s − 0.687·33-s − 0.610·35-s − 1.05·37-s + 0.817·39-s − 0.00492·41-s + 0.301·43-s − 0.149·45-s + 0.576·47-s + 0.862·49-s + 0.583·51-s − 0.392·53-s + 0.532·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.175945766\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.175945766\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
good | 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 + 3.95T + 11T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 19 | \( 1 - 2.79T + 19T^{2} \) |
| 23 | \( 1 - 3.13T + 23T^{2} \) |
| 29 | \( 1 + 1.00T + 29T^{2} \) |
| 31 | \( 1 - 2.03T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 + 0.0315T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 + 3.52T + 59T^{2} \) |
| 61 | \( 1 - 3.00T + 61T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 - 4.27T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 + 1.02T + 83T^{2} \) |
| 89 | \( 1 - 3.28T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.936822532755427346330894023441, −7.44955522338675951050317351312, −6.56416673092618137146169606106, −5.43407989881119587217140327234, −5.17680896662814434068879624140, −4.20849156299778503152722493078, −3.48220474653029176536943060550, −2.75916564878224707979032789946, −1.70394726455677187179456561056, −0.924196896347061758655866552655,
0.924196896347061758655866552655, 1.70394726455677187179456561056, 2.75916564878224707979032789946, 3.48220474653029176536943060550, 4.20849156299778503152722493078, 5.17680896662814434068879624140, 5.43407989881119587217140327234, 6.56416673092618137146169606106, 7.44955522338675951050317351312, 7.936822532755427346330894023441